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Fine aspects of pluripotential theory

El Marzguioui, S.

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Fine Aspects of Pluripotential Theory Said El M

arzguioui

Fine Aspects of Pluripotential Theory

Said El Marzguioui

ISBN: 978-90-9023886-9

Uitnodiging

Voor het bijwonen van de openbare

verdediging van mijn proefschrift

Fine Aspects of Pluripotential

Theory

op woensdag 18 februari 2009 om 14:00 uur aan de Universiteit van Amsterdam in de

Agnietenkapel, Oudezijds Voorburgwal

231 1012 EZ Amsterdam.

Na afloop van de promotieplechtigheid is

er een receptie.

Said El Marzguioui

[email protected]

Fine Aspects of Pluripotential Theory Said El M

arzguioui


Said El Marzguioui

ISBN: 978-90-9023886-9

Uitnodiging

Voor het bijwonen van de openbare

verdediging van mijn proefschrift

Fine Aspects of Pluripotential

Theory

op woensdag 18 februari 2009 om 14:00 uur aan de Universiteit van Amsterdam in de

Agnietenkapel, Oudezijds Voorburgwal

231 1012 EZ Amsterdam.

Na afloop van de promotieplechtigheid is

er een receptie.

Said El Marzguioui

[email protected]

In-RIP-Color-ID-90_Agfa.eps


Said El Marzguioui


Academisch Proefschrift

ter verkrijging van de graad van doctor

aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus

prof. dr. D.C. van den Boom

ten overstaan van een door het

college voor promoties ingestelde commissie,

in het openbaar te verdedigen in de Agnietenkapel

op woensdag 18 februari 2009, te 14:00 uur

door

Said El Marzguioui

geboren te Al Hoceima, Marokko

Promotiecommissie:

Promotor: Prof. dr. J.J.O.O. Wiegerinck

Overige leden: Prof. dr. A. Doelmandr. A. EdigarianProf. dr. E.M. OpdamProf. dr. R.P. StevensonProf. dr. A. Zeriahi

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

THOMAS STIELTJES INSTITUTE

FOR MATHEMATICS

Preface

From its origins in 19th century physics, potential theory has developed into anextensive field of research encompassing concepts like harmonic and subharmonicfunctions, Dirichlet problem, Green functions, and capacities.

In the past century the subject has undergone a very rapid development, andimportant connections to many other mathematical branches were discovered. Asexamples, let us at least mention that potential theory is intimately related tocomplex analysis, probability theory, and the theory of partial differential equa-tions.

Some of these interrelations were further elaborated and have gradually led tothe emergence of new disciplines, largely inspired by potential theory. Examplesare probabilistic potential theory, parabolic potential theory, axiomatic potentialtheory, pluripotential theory and fine potential theory, just to name a few.

Fine potential theory has its roots in the classical Cartan fine topology, andwas mainly developed by Fuglede around 1972. It can be described as the theoryof harmonic and subharmonic functions on open sets with respect to the finetopology. Very soon, this newly developed theory appeared to be useful and hasbrought more precisions and understanding of some classical results in potentialtheory. And more importantly, fine potential theory played a prominent role inthe creation of the theory of finely holomorphic functions. This new conceptof holomorphic functions turned out to be a quite natural generalization of theBorel’s old concept of monogenic functions, which was largely neglected for almosta century.

In this thesis fine potential theory will be used as a tool to treat several prob-lems in pluripotential theory, which on the face have nothing to do with the finepotential theory. An introductory chapter (Chapter 2) is therefore devoted to aquite detailed exposition, though without proofs, of fine potential theory and thetheory of finely holomorphic functions.

Pluripotential theory can be briefly described as the study of plurisubharmonicfunctions and their properties. It relates to holomorphic functions of several com-plex variables just as classical potential theory of R2 = C relates to holomorphicfunctions of one variable.

The study of pluripolar hulls, which will occupy a considerable part of thisthesis, is an example of a new and rapidly developing part of pluripotential theorywithout classical counterpart. Only three years ago Edlund and Joricke [44] dis-

5

6

covered that the theory of finely holomorphic functions can be fruitfully appliedto explain the behavior of pluripolar hulls of graphs of holomorphic functions.This unexpected connection, which actually was implicit in earlier work of Edigar-ian and Wiegerinck [39], raised a whole series of interesting problems, which arestudied in this thesis.

The important role which fine potential theory, and the theory of finely holo-morphic function play in the study of pluripolar hulls, has led us to introduce andstudy the concept of finely plurisubharmonic functions. These functions are theanalogues of plurisubharmonic functions in the so-called pluri-finely open sets. Afirst attempt in this direction was made by El Kadiri [47], see also Fuglede [65, 69].But it was necessary to first overcome some problems about the pluri-fine topologybefore developing “fine pluripotential theory”, or the theory of finely holomorphicfunctions of several variables.

Therefore, we will start in this thesis with a thorough study of the pluri-finetopology, with much focus on connectedness properties. After establishing pleasantproperties of this topology in chapter 3, we turn to study finely plurisubharmonicfunctions. This is done in the same spirit as Fuglede’s finely subharmonic functions.See Chapter 4. It turns out that a rich theory of finely plurisubharmonic functionscan be developed. In fact it will be proved that most fundamental results on finelysubharmonic functions have a counterpart in the finely plurisubharmonic setting.These results will subsequently yield precise information about pluripolar hulls.See Section 1.5 for a complete description of the contents of this thesis.

Acknowledgments

I am greatly indebted to my supervisor (and advisor) Professor Jan Wiegerinckfor his collaboration and excellent guidance. Despite many responsibilities, underwhich the task to run the Korteweg-de Vries institute (KDVI), he was/is alwaysthere for discussions and advice. When new ideas were discovered, there was con-tact even during his vacations or the short stays of one of us at other universities.

When I discussed the possibilities of a PhD project with Jan in 2004, he sug-gested to study the pluri-fine topology and to develop the theory of finely pluri-subharmonic functions. This sounded to me hard and perhaps unworkable project.But what was even inconceivable is that this would help us to understand pluri-polar hulls. Gradually, the plan turned out to be promising, and was eventuallysuccessfully completed. I immensely admire Jan’s strategic approach to problemsand his deep mathematical insight.

Professor Tom Koornwinder was very kind to act as my supervisor during thefirst year. I would like to express my sincere gratitude to him too and thank himfor taking part in the committee.

Next I would like to express my admiration and deep gratitude to ProfessorAhmed Zeriahi who initiated me to pluripotential theory. The courses I took fromhim at the university of Rabat (Morocco) were fundamental for my mathematicalcareer. I am also grateful for his invitations, supervision and the support he gaveme during my stay at Laboratoire Emile Picard (Universite Paul Sabatier) in the

7

period 2001-2002. The hospitality of this institution and the kindness of its staffhave made my time enjoyable.

Also I would like to thank Professor Bent Fuglede for the invitation to themathematics department at Copenhagen university. During a tow-weeks stay wehad very interesting discussions, and he explained to me several results in finepotential theory. I am also indebted to him for the illuminating and fruitfulcorrespondences, and for keeping his interest in my work.

Professor Armen Edigarian was very kind to invite me to the institute of Math-ematics of Jagiellonian University. A visit which was very fruitful. I also wouldlike to thank him for his collaboration which resulted in writing the joint paper[41].

Furthermore I thank my colleague Tomas Edlund. The collaboration with himhas not only resulted in writing a joint paper (Chapter 6), but it was also thebeginning of an excellent friendship.

Thanks also to Guus Balkema for an illuminating discussion concerning Lemma3.4.5.

Also I would like to thank all the promotion committee members for the carefulreading of the manuscript.

The excellent working conditions and the nice atmosphere at the KDV institutehave made the last four years an enjoyable experience. Of course this wouldnot have been possible without very kind friends, colleagues and staff. Thanksto everybody, but in particular Abdelghafour, Evelien, Hicham, Michel, Phyllis,Ramon, Rene and Sjors.

Finally, I am indebted to my parents and brother Hossein for their support inmany ways. My beloved wife Ouarda and my dear children Redouan and Leilahave being so patient when yet another weekend was spent doing mathematics.This is more than enough reason to dedicate this thesis to them.

Contents

Preface 5

Acknowledgments 6

1 Introduction 111.1 Subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Plurisubharmonic Functions . . . . . . . . . . . . . . . . . . . . . . 131.3 Pluripolar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Complete Pluripolar Sets and Thinness . . . . . . . . . . . . . . . 171.5 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Fine Potential Theory 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 The Fine Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Finely Subharmonic Functions . . . . . . . . . . . . . . . . . . . . 352.4 Finely Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . 412.5 Borel’s Monogenic Functions . . . . . . . . . . . . . . . . . . . . . 45

3 The Pluri-fine Topology 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Pluri-thin Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Local Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Further Results on Connectedness . . . . . . . . . . . . . . . . . . 54

4 Finely Plurisubharmonic Functions 614.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Finely Plurisubharmonic Functions . . . . . . . . . . . . . . . . . . 624.3 Continuity of Finely PSH Functions . . . . . . . . . . . . . . . . . 644.4 F-Pluripolar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Finely Holomorphic Functions and Pluripolar Hulls 695.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3 Graphs of Finely Holomorphic Functions . . . . . . . . . . . . . . . 71

9

10 Contents

5.4 Application to Pluripolar Hulls . . . . . . . . . . . . . . . . . . . . 745.5 Concluding Remarks and Open Questions . . . . . . . . . . . . . . 76

6 Fine Analytic Structure 796.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Wermer’s Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3 Proof of Theorem 6.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 836.4 Final Remarks and Open Problems . . . . . . . . . . . . . . . . . . 88

7 Examples and Open Questions 917.1 h-Hausdorff Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2 Borel Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3 Unsolved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Bibliography 99

Index 108

Samenvatting 111

Curriculum Vitae 113

Chapter 1

Introduction

In this introductory chapter we briefly review some properties of plurisubharmonicfunctions and pluripolar sets. Moreover we give a quite detailed description of thecontents of this thesis in Section 1.5.

1.1 Subharmonic Functions

From a historical perspective subharmonic functions appeared first in the remark-able paper of Friedrich Hartogs [74]. However, Hartogs did not give a name tothis new class of functions. It was F. Riesz [119] who first used the french termsubharmonique and started a systematic study of these functions.

In this section we recall some basic properties of subharmonic functions in Rn.For a complete treatment of the subject we refer to [2, 36, 76, 77, 82].

Definition 1.1.1. Let Ω ⊂ Rn be a domain. A function ϕ : Ω −→ [−∞,+∞[ issaid to be subharmonic in Ω ifa) ϕ is upper semi-continuous, i.e, lim supx→a ϕ(x) ≤ ϕ(a) for each point a ∈ Ω;b) for every point a ∈ Ω and every r > 0 such that B(a, r) ⊂ Ω, the value ϕ(a)does not exceed the average value of ϕ on the sphere ∂B(a, r):

ϕ(a) ≤1

σn(∂B(a, r))

∫

∂B(a,r)

ϕ(x)dσn, (1.1.1)

where σn is the Lebesgue measure on the sphere ∂B(a, r).

We shall let SH(Ω) designate the set of subharmonic functions on Ω. Since weare mainly interested in the complex case, i.e, Cn, n ≥ 1, Examples of subharmonicfunctions are postponed to the next section. Also, in order not to say everythingtwice we will only provide here a brief discussion and move on to the next section.

Condition a) in the above definition amounts to x ∈ Ω : ϕ(x) < c beingopen for each c ∈ R. Hence, by a simple covering argument, one can show thatϕ attains its maximum on each compact subset K ⊂ Ω. Moreover, the submean

11

12 Chapter 1. Introduction

value property (1.1.1) forces subharmonic functions to “attain” their maximum onthe boundary. Explicitly, the following maximum principle holds .

Theorem 1.1.2. If Ω is a bounded connected open subset of Rn, and if ϕ ∈ SH(Ω),then either ϕ is constant or, for each x ∈ Ω,

ϕ(x) < supz∈∂Ω

lim supy→z,y∈Ω

ϕ(y)

. (1.1.2)

The following properties follow immediately from the definition of subharmonicfunctions:1) SH(Ω) is a convex cone, i.e c1u1 + c2u2 ∈ SH(Ω) for any c1, c2 ≥ 0 and anyu1, u2 ∈ SH(Ω).2) If u1, ..., u2 ∈ SH(Ω), then the function u(x) = max(u1(x), ..., u2(x)) belongsalso to SH(Ω).3) The limit of a uniformly convergent or monotonically decreasing sequence ofsubharmonic functions is subharmonic.

The upper semi-continuity in Definition 1.1.1 is the appropriate condition forcertain key results (e.g property 3) above). Noteworthy is that although subhar-monic function can be highly discontinuous in general, they possess some interest-ing continuity properties, see Theorem 2.3.16.

It can be proved that replacing inequality (1.1.1) in the above definition by thefollowing one, yields an equivalent definition

ϕ(a) ≤1

λn(B(a, r))

∫

B(a,r)

ϕ(x)dλn, (1.1.3)

Here dλn is the Lebesgue measure in Rn. Observe that subharmonic functions areallowed to take the value −∞. In fact, the sets of the form ϕ = −∞, whereϕ 6≡ −∞, are called polar and they play an important role in potential theory andcomplex analysis. Observe that using (1.1.3), one can easily prove the following.

Theorem 1.1.3. If Ω ⊆ Rn is connected and ϕ ∈ SH(Ω), then either ϕ ≡ −∞ orϕ ∈ L1

loc(Ω).

Here L1loc(Ω) denotes the family of locally Lebesgue integrable functions in Ω.

As a consequence of Theorem 1.1.3, polar sets have Lebesgue measure zero. Avery particular class of polar subsets of R2n, the so-called pluripolar sets, will beintroduced and discussed in section 1.3. In fact, pluripolar sets will be a centraltopic in the next chapters.

The next theorem characterizes subharmonic functions in terms of the Laplaceoperator.

Theorem 1.1.4. If Ω ⊆ Rn is open and ϕ ∈ SH(Ω) such that ϕ 6≡ −∞, then 4ϕcomputed in the sense of distribution theory is a positive measure. Conversely, ifv ∈ L1

loc(Ω) is such that 4v ≥ 0 in Ω in the sense of distributions, then thereexists a function ϕ ∈ SH(Ω) such that ϕ = v almost everywhere in Ω.

1.2. Plurisubharmonic Functions 13

The function ϕ in the second part of the above theorem is obtained as a limitof convolutions of v with the standard smoothing kernels. We must also point outthat ϕ need not coincide everywhere with v: for example, if v is the characteristicfunction of a compact subset K of Ω of measure 0, then ϕ ≡ 0.

In potential theory one often has to take upper envelopes of subharmonic func-tions. This method, called balayage or swepping out, goes back to H. Poincare[111] and was systematically studied by Brelot [15]. The method of balayage hasbeen widely applied and successfully used in different areas. For example, the har-monic measure as a ”solution” of a particular Dirichlet problem is obtained via thismethod, and it serves as a decisive tool with applications ranging from complexanalysis to probability theory. This method of balayage partly owes its success tothe following key property of subharmonic functions known as the fundamentalconvergence theorem of Cartan:

Theorem 1.1.5. If uα, α ∈ A ⊂ SH(Ω) is locally bounded above on Ω, and ifU(x) = supα∈A uα(x), then the function

U∗(x) := lim supy→x

U(y)

is again subharmonic on Ω. Moreover, the set U < U ∗ is a polar subset of Ω.

The function U∗ is called the upper semi-continuous regularization of U . Wenote in passing that in axiomatic potential theory the assertion of Theorem 1.1.5is sometimes taken as an axiom [27, page 100].

1.2 Plurisubharmonic Functions

The aforementioned paper of Hartogs [74] from 1908 contains some fundamentalresults on the theory of holomorphic functions of several variables, including thestriking fact that when n > 1 holomorphic functions can not have isolated singu-larities nor isolated zeros. More precisely, on Cn for n > 1, any analytic functionF defined on the complement of a compact set K extends (necessarily uniquely)to an analytic function on Cn. Thus, whereas every plane domain carries noncon-tinuable holomorphic function (i.e, a domain of holomorphy), there are domains inCn, n > 1, that do not enjoy this property. This Hartogs extension phenomenonhas led to the question of characterizing the domains of holomorphy.

E. E. Levi discovered around 1912 that a domain of holomorphy must satisfya certain convexity condition known as pseudoconvexity. He then asked if thiscondition characterizes domains of holomorphy. This question turned out to bevery difficult and became known as the Levi problem. See Kiselman’s survey [79].

In the course of studying the Levi problem Kiyoshi Oka [108] introduced thenotion of plurisubharmonic function who then cracked the problem in 1942, firstin dimension two and later in all dimensions, see [79]. Lelong [84] independentlyintroduced plurisubharmonic functions in the same year, and subsequently studiedthem in great detail. Nowadays, the theory of plurisubharmonic functions, labelled


pluripotential theory, has become a major field of research with a wide variety ofapplications. See e.g. [90] and [103]. Klimek’s monograph [80] provides a goodintroduction to the theory of plurisubharmonic functions. See also [77].

Definition 1.2.1. Let Ω ⊂ Cn be a domain. A function ϕ : Ω −→ [−∞,+∞[ issaid to be plurisubharmonic (psh in short) in Ω ifa) ϕ is upper semi-continuousb) for every complex line L ⊂ Cn, ϕ|Ω∩L is subharmonic on Ω ∩ L

If ϕ and −ϕ are plurisubharmonic, we say that ϕ is pluriharmonic in Ω. Theset of plurisubharmonic functions on Ω is denoted by PSH(Ω) . Note that if Ω isa plane domain, i.e, n = 1, then PSH(Ω) = SH(Ω)

An equivalent way of stating property b) is: for all a ∈ Ω, b ∈ Cn, such that

a+ λb : λ ∈ C, |λ| ≤ 1 ⊂ Ω,

we have

ϕ(a) ≤1

2π

∫ 2π

0

ϕ(a+ eiθb)dθ. (1.2.1)

An integration of (1.2.1) over b ∈ ∂B(a, r) shows that (1.1.1) in Definition 1.1.1holds. Hence PSH(Ω) ⊂ SH(Ω) ⊂ L1

loc(Ω).It is immediate that the properties 1), 2) and 3) from Section 1.1 for subhar-

monic functions are valid also for plurisubharmonic ones.If ϕ is twice continuously differentiable in Ω, then ϕ is psh if and only if for

each z ∈ Ω and b ∈ Cn, the Laplacian of C 3 λ 7→ ϕ(z + λb) is nonnegative atλ = 0 (cf. Theorem 1.1.4), i.e,

∂2ϕ

∂λ∂λϕ(z + λb) =

n∑

j,k=1

∂2ϕ

∂zj∂zk(z + λb)bjbk ≥ 0, ∀z ∈ Ω, ∀b ∈ Cn. (1.2.2)

Therefore, ϕ is plurisubharmonic on Ω if and only if the complex Hessian

[

∂2ϕ

∂zj∂zk(z)

]

(1.2.3)

of ϕ is positive semidefinite at every point z ∈ Ω. We should mention that thischaracterization is still valid for arbitrary psh functions, namely, the analogue ofTheorem 1.1.4 with the Laplacian replaced by the “distributional” Hessian matrixholds.

It is well known that convex functions in R2n that are twice continuouslydifferentiable are characterized by the positivity of the real Hessian. Thus pluri-subharmonic functions may be viewed as a generalization of the former. In factwe have the following.

Example 1.2.2. Any continuous convex function ϕ in Ω ⊂ Cn (with respectto the underlying real variables) is plurisubharmonic in Ω. Indeed, it suffices to

1.3. Pluripolar Sets 15

integrate the following inequality with respect to dθ/2π to get the submean value(1.2.1).

ϕ(a) ≤ϕ(a+ eiθb) + ϕ(a− eiθb)

2.

More interesting examples of psh functions include |f | and c log |f | for a con-stant c > 0 and a holomorphic function f in Ω. In fact, this class of functionsgenerates locally the set PSH(Ω) in the following way.

Theorem 1.2.3 (Bremermann [18]). Every function ϕ ∈ PSH(Ω) is givenlocally as

ϕ(z) = lim supw→z

(lim supj→∞

1

jlog |fj(w)|), (1.2.4)

for some sequence fj of holomorphic functions.

It should be mentioned here that for a plane domain Ω, this theorem wasdiscovered by Lelong [83].

For an upper semi-continuous function the second property in Definition 1.2.1says that ϕf should be subharmonic wherever it is defined, for all affine mappingsf : C −→ Cn. It is a remarkable fact that this property implies that ϕ f issubharmonic also for every holomorphic mapping f . Conversely, we have thefollowing

Theorem 1.2.4. A function ϕ : Ω −→ [−∞,+∞[ defined on an open set Ω ⊂ Cn

is plurisubharmonic in Ω if and only if ϕ F is subharmonic in F−1(Ω) for everyC-linear isomorphism F : Cn −→ Cn.

Theorem 1.2.3, and 1.2.4 explain already why psh functions are the right classof subharmonic functions to be studied in complex analysis.

1.3 Pluripolar Sets

A set E ⊂ Cn is called pluripolar if for every point z ∈ E there is an opensubset Ω ⊂ Cn containing z and a function ϕ ∈ PSH(Ω) (ϕ 6≡ −∞) such thatE∩Ω ⊂ ϕ = −∞. These sets were first introduced and studied by Lelong [85, 86]who asked whether this local definition is equivalent to the global one; that is, givena pluripolar set E ⊂ Cn, does there exist a globally defined plurisubharmonicfunction ϕ ∈ PSH(Cn) such that E ⊂ ϕ = −∞? This problem has been calledthe first Lelong problem and was around for more than two decades. An affirmativeanswer was given by Josefson [78] with the use of delicate polynomial estimates.See also [77, page 285-290] for a more transparent version of Josefson’s proof.

Trivial examples of pluripolar sets are the zero sets of holomorphic functions,the intersection of such sets, and real-analytic curves, see [122, page 68-72]. How-ever, despite the close connection (see. Theorem 1.2.3) between plurisubharmonicfunctions and holomorphic ones, pluripolar sets can be very nasty and far more


complicated in structure than complex analytic varieties. In fact, there are ex-amples of pluripolar sets that contain no non-trivial analytic variety and even nofinely analytic curves. See [91] and Chapter 6.

In view of the important role in classical potential theory of the Cartan’s re-sult, cf. Theorem 1.1.5, it is natural to ask whether the corresponding result forplurisubharmonic functions holds; that is, if uα, α ∈ A ⊂ PSH(Ω) is a familyof plurisubharmonic functions which are locally uniformly bounded above on Ω,and if U(z) = supα∈A uα(z), is then the set U < U∗ pluripolar? This problem(called the second Lelong problem) appeared first in the work of Lelong [87, 88]who called these kind of sets negligeables and studied them thoroughly. However,although Lelong succeeded to settle the problem in the particular case where U ∗(z)is pluriharmonic, cf. [88, proposition 7], the problem turned out to be very hardin the general case, and had to await for further development of pluripotentialtheory.

In their pioneering paper [4], Bedford and Taylor introduced and studied theMonge-Ampere capacity. And they showed in particular that the zero sets of thiscapacity are precisely the pluripolar sets. This provided the missing instrumentfor the study of pluripolar sets, a new proof of Josefson’s theorem, and a solutionto the second Lelong problem.

Since PSH(Ω) ⊂ SH(Ω), pluripolar sets are polar, and hence have Lebesguemeasure zero in view of Theorem 1.1.3. Moreover, pluripolar sets should be con-ceived as even ”smaller” since they have zero Monge-Ampere capacity. On theother hand, in an answer to a question by Bedford, Diederich and Fornaess [33]constructed a C∞-smooth, real, closed curve γ in C2 which is not pluripolar. Inparticular, the Hausdorff dimension of γ is equal to 1. This reveals a substantialdifference between the notion of pluripolarity in Cn and that of polarity in R2n.Indeed, it is a well known result, cf. [2, page 156-159], that the Hausdorff dimen-sion of a polar set P ⊂ Rk, k ≥ 3 is smaller or equal to k − 2. In the oppositedirection, every set P ⊂ Rk, k ≥ 3 of Hausdorff dimension strictly smaller thank − 2 is necessarily a polar set. It is noteworthy that Carleman [20, Theorem 1]has constructed a non-polar set E ⊂ C which has Hausdorff dimension zero. Theproduct set E×E is evidently non-pluripolar sets in C2, and again with Hausdorffdimension zero, see also [6, 81].

Pluripolar sets are an essential ingredient in the study of the growth of entirefunctions in Cn, n > 1; a well known result asserts that the exceptional set ofthe partial order of growth of an entire function is pluripolar, see [90]. In theopposite direction, Zeriahi [139] showed that every complete pluripolar set (seeSection 1.4) E ⊂ Cn of type Fσ is the exceptional set of the partial order of thegrowth of some entire function in Cn. In approximation theory, Sadullaev [123]proved that every holomorphic function on a domain of holomorphy D ⊂ Cn israpidly approximable by rational functions if and only if the complement of D isa pluripolar set. Furthermore, pluripolar sets are usually removable for plurisub-harmonic functions, holomorphic functions and closed positive currents. However,for a pluripolar set to be removable, it is sometimes necessary to require that it iscomplete. See [51, 126] and the next section.

1.4. Complete Pluripolar Sets and Thinness 17

1.4 Complete Pluripolar Sets and Thinness

A set E ⊂ Cn is pluri-thin at a point a ∈ Cn if and only if either a is not a limitpoint of E or there exist r > 0 and a plurisubharmonic function ϕ on B(0, r) suchthat

lim supz→a,z∈E\a

ϕ(z) < ϕ(a). (1.4.1)

We say that E is pluri-thin if it is pluri-thin at all of its points. In such acase E is negligible, cf. [4, Theorem 7.1], hence pluripolar, cf. [22, Theorem 2.1].In contrast to the situation in classical potential theory (see Example 2.2.13),pluripolar sets are not necessarily pluri-thin as is shown by the following simpleexample.

Example 1.4.1. If E = (z, w) ∈ C2 : |z| < 1, w = 0, then E is pluripolar butE is not pluri-thin at any of its points.

Observe that the set E in this example is contained in the complex line w = 0.Thus, one just uses property b) in Definition 1.2.1 together with the fact that asubharmonic function on w = 0 can not ”jump” at points of E, see Subsection2.2.2. Notice that this argument rests on the set E being big in some complexanalytic variety. There is a criterion of pluri-thinness involving pluriharmonicmeasure, cf. [5, page 228], which has limited applications, since the pluriharmonicmeasure is not easily computed. Thus usually, it is hard to determine whether aset E is a pluri-thin at a given point z. See Theorem 1.4.6 and Theorem 1.4.7.

We shall now confine the discussion to pluripolar sets. The question: “When isa pluripolar set E pluri-thin at a given point z”? hides a whole series of problems.We need some definitions.

Definition 1.4.2. Let Ω be an open subset of Cn. A pluripolar set E ⊂ Ω iscalled complete pluripolar in Ω if there exists ϕ ∈ PSH(Ω) such that

E = z ∈ Ω : ϕ(z) = −∞.

In view of the upper semi-continuity of psh functions, a complete pluripolarset must be a Gδ set. In the classical potential theory of Rk, a well known resultof Choquet [24], see also [82, Theorem 3.1], asserts that a polar set E ⊂ Rk iscomplete polar in Rk if and only if E is polar and a Gδ set. The proof of this isheavily based on the Riesz decomposition theorem, which ensures that every polarset E ⊂ Rk is contained in the infinity set of some potential. The situation forpluripolar sets is far more complicated. There are easy examples of Gδ pluripolarsets that are not complete in Cn. Moreover, since plurisubharmonic functionsare assumed to be subharmonic when restricted to lower dimensional complexmanifolds, the pluripolar sets will obviously exhibit a propagation behavior as thefollowing simple example illustrates. See [135].

Example 1.4.3. Let E = (z, w) ∈ C2 : |z| = 1, w = 0. The functionlog |w| ∈ PSH(C2) equals −∞ on E, so E is pluripolar. Moreover E is a Gδ .


However, every function ϕ ∈ PSH(C2) which is equal to −∞ on E will be equal to−∞ on the larger set C×0. Indeed, the function ψ(z) = ϕ(z, 0) is subharmonic inC and is equal to −∞ on the circle |z| = 1. By the maximum principle ψ(z) = −∞for every |z| ≤ 1. Theorem 1.1.3 implies now that ψ(z) ≡ −∞.

Trivial examples of complete pluripolar sets in Cn are graphs of entire functions.Indeed, if f(z) is a holomorphic function in Cn−1, then the function (z, w) 7→log |w− f(z)| is plurisubharmonic in Cn and assumes the value −∞ on the graph

Γf (Cn−1) = (z, f(z)) : z ∈ Cn−1, (1.4.2)

but nowhere else. This means that Γf (Cn−1) is a complete pluripolar set in Cn.Less obvious examples of complete pluripolar sets will be given below.

The study of completeness of pluripolar sets has naturally led to the followingconcept.

Definition 1.4.4. Let E be a pluripolar set in Cn. The pluripolar hull of Erelative to an open subset Ω of Cn is the set

E∗Ω = z ∈ Ω : for all ϕ ∈ PSH(Ω) : ϕ|E = −∞ =⇒ ϕ(z) = −∞.

For example, it is clear that the pluripolar hull of the compact set E in example1.4.3 is the set C×0. An example of a compact pluripolar set K ⊂ C2 such thatK∗

Cis dense in C2 may be found in [6]. Let us remark that (E ∪ F )∗Ω = E∗

Ω ∪ F ∗Ω.

Moreover the ”star” operation is idempotent; that is, (E∗Ω)∗Ω = E∗

Ω.If a point z ∈ Ω does not belong to E∗

Ω, then E is evidently pluri-thin at z.Whether the converse holds is not known.

Clearly, if E is complete pluripolar in Ω, then E is a Gδ and E = E∗Ω. It

is not known whether the converse is true. However, in [139] Zeriahi proved thefollowing.

Theorem 1.4.5. Let E be a pluripolar subset of a pseudoconvex domain Ω ⊂ Cn

and suppose that E = E∗Ω. If E is a Gδ and an Fσ set, then E is complete

pluripolar in Ω.

The notion of the pluripolar hull was first introduced and studied by Zeriahiin [139]. The paper [93] of Levenberg and Poletsky contains a more detailed studyof this concept, and a proof of the following theorem.

Theorem 1.4.6. Let E = (t, tα) ∈ C2 : 0 < t ≤ 1 with α irrational. ThenE∗

C2 = (z, zα) ∈ C2 : z 6= 0, where zα runs over all possible values. In particular,E is pluri-thin at the origin.

This solved an old problem of Sadullaev [122], who observed already that theset E is not pluri-thin at the origin when α is rational. This is because the curveE is real analytic at t = 0 in this case. See [122, Proposition 4.1]. The set E∗

C2 inTheorem 1.4.6 is not a Gδ . Hence it is not complete.

In the same direction Wiegerinck [133] proved the following theorem whichanswers a second question of Sadullaev [122].

1.4. Complete Pluripolar Sets and Thinness 19

Theorem 1.4.7. Let E = (t, e−1/t) ∈ C2 : 0 < t ≤ 1. Then E∗C2 =

(z, e−1/z) ∈ C2 : z 6= 0. In particular, E is pluri-thin at the origin.

These two theorems should be compared with Theorem 2.2.15. Also, observethat the graph (E∗

C2) of e−1/z over C\0 is complete pluripolar in C2 in view ofZeriahi’s theorem. In [133], it was actually proved that Theorem 1.4.7 remainsvalid if e−1/z is replaced by certain other holomorphic functions with an essentialsingularity at 0. However, soon after this, Wiegerinck [134] discovered the followinggeneral result. See also [38].

Theorem 1.4.8. Suppose that D is a domain in C and A a sequence of points inD without density point in D. Let f be holomorphic on D\A, and not extendibleover A. Let E denote the graph of f in D\A× C. Then E is complete pluripolarin D × C.

It should be mentioned at this point that the above results are related toquestions of analytic continuation of holomorphic functions in Cn. To illustratethis connection let us quote here the following result, known as the strong “disc”theorem, of Bremermann [19]. See also [54].

Theorem 1.4.9. Let z(t): [0, 1] −→ Cn be a curve contained in a complex lineL ⊂ Cn, and let the function f(z, w), z ∈ Cn, w ∈ C, be holomorphic in a domainΩ ⊂ Cn, containing the set D(t) = (z, w) : z = z(t), |w| < ρ for 0 < t < 1 andρ > 0. Suppose that the function f(z(0), w) is holomorphic at at least one pointof the set D(0) = (z, w) : z = z(0), |w| < ρ. Then f(z(0), w) is holomorphiceverywhere in D(0).

The real analytic curves in Theorem 1.4.6 and Theorem 1.4.7 are not analyti-cally continuable at the origin. This shows that non-analytic continuation mightbe the obstruction to the propagation of pluripolar sets. This point of view hasbeen tested on graphs of holomorphic functions too. Indeed, inspired by a funda-mental example of Sadullaev [122], Poletsky, Levenberg and Martin showed thatthe graphs of many lacunary series are complete pluripolar in C2. This has ledthem to conjecture that if f is a holomorphic function, that is defined on its max-imal domain of existence D ⊂ C, then the graph Γf of f is complete pluripolar inC2.

A break through was achieved by Edigarian and Wiegerinck [39] by showingthat the above conjecture fails. Their counterexample is given by a function

f(z) =

∞∑

j=1

cjz − aj

, (1.4.3)

where an∞n=1 is a countable and dense sequence, say in the boundary ∂D of theunit disk D. Choosing cj to be very rapidly decreasing to 0, and using very preciseharmonic measure estimates, they showed that there is a point a ∈ ∂D for which(a, f(a)) ∈ (Γf (D))∗

C2 . Here Γf (D) is the graph of f over D.


This result gave a substantial impulse to the study of the pluripolar hullsof graphs of holomorphic functions. Siciak [125] observed that the function fabove has a pseudocontinuation f across almost every point of the unit circle, andshowed that the graph Γf of f is contained (Γf (D))∗

C2 . In the opposite direction,he gave an example of a holomorphic function g in the unit disk which has nopseudocontinuation, while (Γg(D))∗

C2 6= Γg(D).In [44], Edlund and Joricke observed subsequently that the function f of Edi-

garian and Wiegerinck is actually finely holomorphic (see, Section 2.4) in a setlarger than the unit disk. In other words, the function f has a finely holomorphicextension f beyond the unit disk. Similarly to the above quoted result of Siciak,they showed that (Γf (D))∗

C2 contains a part of the graph of the finely analytic

extension f , see Theorem 5.1.1. Thus, for a holomorphic function g, say in theunit disk D, fine analytic continuation is a sufficient condition for (Γg(D))∗

C2 to bestrictly larger than (Γg(D)). Whether this is also a necessary condition remainsunknown. It is however worth mentioning that a partial result in this directionwas obtained by T. Edlund in his thesis [43]. See also [42] for a related result.

One of our goals in this thesis is to investigate to what extent the theory offinely holomorphic functions of one and several variables is related to pluripolarhulls. However, no theory of finely holomorphic functions of several variablesexists. To develop it one must start with a study of the plurifine topology andfinely plurisubharmonic functions. Thus we arrive at the plan of this thesis below.

1.5 Overview of the Thesis

Many of the results in this thesis were obtained in collaboration with Jan Wiegerinck.The contents is based on the following papers.I. El Marzguioui, S. Wiegerinck, J.: The pluri-fine topology is locally connected.Potential Anal 25 (2006), no. 3, 283–288.II. El Marzguioui, S., Wiegerinck, J.: Connectedness in the pluri-fine topology,to appear in Proceedings of the Conference on Functional Analysis and ComplexAnalysis Edited by: A. Aytuna, R. Meise, T. Terzioglu, and D. Vogt.III. El Marzguioui, S., Wiegerinck, J.: Continuity Properties of Finely Plurisub-harmonic Functions and Pluripolarity, preprint 2008.IV. Edigarian A., El Marzguioui, S., Wiegerinck, J.,: The image of a finely holo-morphic map is pluripolar, Preprint. arXiv: math/0701136.V. Edlund, T., El Marzguioui, S.: Pluripolar hulls and fine analytic structure, toappear in Indagationes Mathematicae.

To get a quick idea about the contents of this thesis, it is illuminating tonotice that all our results were obtained in the course of studying the followingtwo problems.Problem 1. Study the pluri-fine topology in Cn and develop the theory of finelyplurisubharmonic and pluri-finely holomorphic functions.

Having some grip on Problem 1 allows us to attack Problem 2Problem 2. Study the pluripolar hulls and understand their structure.

1.5. Overview of the Thesis 21

Problem 1 is also related to the study of the complex Monge-Ampere operator[5], and the more precise pointwise behavior of psh functions.

Let us state the following two problems which were presented at the ProblemSession during a conference at Hanstholm, cf. [69, page 195-196]

Question 1. (Fuglede) Is the pluri-fine topology locally connected?

Question 2. (P. M. Gauthier) Let K be a compact subset of Cn and let PS(K)denote the set of functions onK which can be uniformly approximated by functionswhich are continuous and plurisubharmonic on (neighborhoods of) K. Give acharacterization of functions in PS(K).

In the case n = 1, a rich theory of finely subharmonic (resp. finely holomorphic)functions was developed in the period 1969-1981, and Questions 1 and 2 weresolved.

In Chapter 2 we will survey the development of fine potential theory, and finelyholomorphic functions. This will serve as a source of inspiration and a model forour study of Problem 1. It may be convenient for the reader as well. Chapters 3-7contain the original work of the thesis.

1.5.1 Fine Pluripotential Theory

The pluri-fine topology on an open set Ω ⊆ Cn is the coarsest topology on Ω thatmakes all plurisubharmonic functions on Ω continuous. We shall use the notationsfrom Subsection 3.1.1.

In Chapter 3 we show that the pluri-fine topology has pleasant connectednessproperties. It is locally connected (Theorem 3.3.4), and a usual open set U ⊂ Cn

is F-connected if and only if U is connected in the usual topology. This gives anaffirmative answer to Question 1 above. See also [6, 65].

As a consequence of the local connectedness and the so-called quasi-Lindelofproperty, cf. Theorem 3.2.5, it follows quite easily that every F-open set has acountable number of F-connected components. Moreover, these components areF-open. Our proof of the local connectedness relies ultimately on two properties offinely subharmonic functions in C: Theorem 2.3.9, and Lemma 2.3.13 (the gluinglemma).

Next we introduce an analogue of plurisubharmonic functions on F-domains.A natural definition is as follows: a real function f on an F-open set U ⊆ Cn

is said to be finely plurisubharmonic in U (or F-plurisubharmonic) if it is uppersemicontinuous in the pluri-fine topology and finely subharmonic on each complexline where it is defined. The key ingredients in the study of these functions arecontained in the following result which is proved in Chapter 3.

Theorem 1.5.1. Let U ⊆ Cn be an F-open subset and let a ∈ U . Then thereexists a constant κ = κ(U, a) and an F-neighborhood V ⊂ U of a with the propertythat for any complex line L through z ∈ V the F-component Cz,L of the F-openset U ∩ L that contains z, contains a circle about z with radius at least κ.

The proof of Theorem 1.5.1 relies on a classical harmonic measure estimate of


A. Beurling and R. Nevanlinna, and leads to a second proof of the local connect-edness of the pluri-fine topology. See Theorem 3.4.10 and Remark 3.3.7.

In Chapter 4 we study F-plurisubharmonic functions. The first non-trivialresult reads as follows.

Theorem 1.5.2. Let f be an F-plurisubharmonic function on an F-domain Ω.If f = −∞ on an F-open subset U of Ω, then f ≡ −∞.

Theorem 1.5.2 has interesting applications. Firstly, the F-plurisubharmonicfunctions satisfy the maximum principle in a sense made precise in Theorem 4.2.6.Secondly, pluripolar sets don’t separate F-domains, cf. Theorem 4.2.4. Finally,Theorem 1.5.2 can be applied to study pluripolar hulls, cf. [49].

Furthermore we prove in Section 4.3 that every bounded F-plurisubharmonicfunctions can be F-locally written as the difference of two usual plurisubharmonicfunctions, cf. Theorem 4.3.1. As a consequence of this, we show that F-pluri-subharmonic functions are pluri-finely continuous (not just pluri-finely upper semi-continuous, by definition). This means that there is no larger “plurifine-plurifine”topology. See Theorem 2.3.12 for the finely subharmonic case.

Another consequence of this F-local decomposition of F-plurisubharmonicfunctions and Theorem 1.5.2 is the following Theorem, which gives an affirma-tive answer to a question in [47].

Theorem 1.5.3. Let f be an F-plurisubharmonic function on an F-domain Ωsuch that f 6≡ −∞. Then the set f = −∞ is pluripolar subset of Ω.

Note that since F-plurisubharmonic functions in a planar domain are finelysubharmonic, our proof of the above theorem provides a second proof of Theorem2.3.14.

Theorem 1.5.3 will be applied to questions about pluripolar hulls. This isdiscussed in the next subsection.

1.5.2 Applications to Pluripolar Hulls

Here we shall outline the progress we have made so far in studying Problem 2.To formulate the main result of Chapter 5 we need the following definition.

Definition 1.5.4. Let U ⊆ Cn be F-open. A function f : U −→ C is said to beF-holomorphic if every point of U has a compact F-neighborhood K ⊆ U suchthat the restriction f |K belongs to H(K).

As was pointed out at the end of the preceding section, Theorem 1.5.3 allowsus to prove the following result, cf. Theorem 5.3.1, and Theorem 5.4.3.

Theorem 1.5.5. Let U ⊂ Cn be an F-domain. Let f be F-holomorphic in U .Then the zero set of f is pluripolar. In particular, the graph of f is also pluripolar.Moreover, if E is a non-pluripolar subset of U , then Γf (U) ⊂ (Γf (E))∗

Cn+1 .

The above theorem generalizes the following result, which was obtained incollaboration with Edigarian and Wiegerinck, cf. Theorem 5.4.1.

1.5. Overview of the Thesis 23

Theorem 1.5.6. Let f : U −→ Cn, f(z) = (f1(z), ..., fn(z)), be a finely holo-morphic map on an F-domain U ⊆ C. Then the image f(U) of U is a pluripolarsubset of Cn. Moreover, if E is a non polar subset of U , then the pluripolar hullof f(E) contains f(U).

It is interesting to observe that the set U in the above theorems need not haveany Euclidean interior points.

It turns out that the arguments in the proof of Theorem 1.5.6 can be equallywell applied to arbitrary pluripolar hulls, rather than the particular case of graphs,see Propositions 5.5.1.

Chapter 6 is joint work with Tomas Edlund. Here we constructed an example,cf. Theorem 6.2.4, of a compact pluripolar set Yδ which hits every graph of a finelyholomorphic function in a polar set, and yet Yδ 6= (Yδ)

∗C2 . This initial example is

further elaborated to construct a pluripolar set with a very large pluripolar hullwithout fine analytic structure. Explicitly, we have the following.

Theorem 1.5.7. For each proper non polar subset S ⊂ C there exists a pluripolarset E ⊂ (S×C) with the property that E∗

C2 contains no fine analytic structure andthe projection of E∗

C2 onto the first coordinate plane equals C.

The set E in the above theorem is a subset of a complete pluripolar setX ⊂ C2,which is constructed in the same spirit as Wermer’s polynomially convex compactset without analytic structure. What is actually used is that X is the graph of ananalytic multifunction.

Finally, in Chapter 7 we illustrate our results by examples and present someopen problems.

Chapter 2

Fine Potential Theory

Since fine potential theory has proven to be a powerful tool in studying the pluri-fine topology and questions related to pluripolar sets, we found it convenient towrite a chapter summarizing the basic properties of finely (sub) harmonic func-tions and finely holomorphic ones. The theory of finely harmonic functions wasdeveloped by Fuglede [56] in the general frame work of abstract harmonic spaces.However, we shall restrict our exposition to the particular case of a Green setΩ ⊂ Rn, i.e, an open set having a Green function. This amounts, in view ofMerberg’s theorem, to Ω being an arbitrary open set if n > 2 or Ω has non-polarcomplement if n = 2. Furthermore, since the case n > 2 is not relevant to the sub-ject of the next chapters, and for the sake of simplicity, we will confine ourselves tothe case where Ω ⊂ C is an open subset of the complex plane with non-polar com-plement. Nevertheless, all the results presented in Sections 2.2 and 2.3.2, unlessotherwise indicated, are valid also in the general case, i.e, n ≥ 2.

2.1 Introduction

Potential theory in the complex plane C can be roughly characterized as the theoryof the Laplace operator

∆ =∂2

∂x2+

∂2

∂y2. (2.1.1)

A function h is said to be harmonic on the open set Ω ⊂ C, if h has continuoussecond partial derivatives on Ω and ∆h(z) = 0 for all z ∈ Ω.

A considerable part of potential theory was developed in the course of studyingthe Dirichlet problem and related questions. The problem may be stated, in itssimplest form, as follows: for a given continuous function f : ∂Ω −→ R, determine,if possible, a harmonic functions h on Ω such that h(z) → f(ζ) as z → ζ for eachζ ∈ ∂Ω. Such a function h is called the solution of the Dirichlet problem on Ω withboundary function f , and the maximum principle guarantees the uniqueness of thesolution if it exists. The set Ω is said to be regular, provided that the Dirichlet

25

26 Chapter 2. Fine Potential Theory

problem has a solutions for all continuous function on ∂Ω. Simple examples ofregular sets are a disk or a half-plane. For these particular sets, the solution iseven given by a nice integral representation involving the Poisson kernel, see [2, 76].On the other hand, there are quite simple examples of non-regular sets. The firstsuch a simple example goes back to S. Zaremba [138].

Example 2.1.1. If Ω = D(0, 1)\0 is the punctured unit disk in C, and f : ∂Ω−→ R is defined by f(0) = 1 and f(ζ) = 0 when |ζ| = 1, then it is an immediateconsequence of the removable singularity theorem and the maximum principle thatthere is no harmonic function on Ω with the preassigned boundary values.

Thus Zaremba discovered that domains with isolated boundary points are notregular. In other words, the Dirichlet problem does not always have a solution forsuch domains. A non trivial example (Lebesgue spine) of a non-regular domain inR3 was given by H. Lebesgue in (1913), cf. [76].

It was Lebesgue who explicitly proposed to ”separate” the investigation ofthe Dirichlet problem into two parts: Firstly to produce a harmonic functiondepending in a way on the given boundary function and then investigate theboundary behavior of the resulting candidate for a solution. The classical Perron-Wiener-Brelot method (PWB-method for short) does associate to each boundaryfunction f such a candidate function (denoted by Hf ) as the supremum of acertain family of subharmonic functions associated to f . See e.g, [2, 76]. Thispowerful method has brought in the concept of a regular boundary point. Aboundary point ζ ∈ ∂Ω is called regular if for any continuous function f on ∂Ω,limz→ζ Hf (z) = f(ζ). Otherwise ζ is called an irregular boundary point. In otherwords, an irregular point of the boundary of a domain Ω is a point at whichthe continuity of the PWB solution may be violated. For instance, the origin inExample 2.1.1 is irregular and the PWB solution is the constant Hf ≡ 0. See [117,chapter 4].

In the first half of the 20th century quite an extensive study of the irregularboundary points was carried out. A major achievement was Wiener’s characteriza-tion of these points in terms of capacity in 1924, cf. Theorem 2.2.16. This famouscriterion of irregularity turned out to be useful and has certainly contributed tofurther development of potential theory. However, in general it is hard to computeor estimate the capacity of a given set, and therefore Wiener’s criterion is not easilyapplicable in some practical situations. The need to find more manageable criteriafor irregular points perhaps motivated Marcel Brelot to introduce the notion ofthinness of a set E ⊂ C at a point x ∈ C. In 1939 he defined E to be thin at a ifeither a is not a limit point of E, or else if there exists a subharmonic function ϕin a neighborhood of a such that

lim supx→a, x∈E\a

ϕ(x) < ϕ(a). (2.1.2)

Observe that since ϕ is upper-semicontinuous, we always have ≤ in (2.1.2).Brelot showed that for a closed set E which contains x, this definition amounts

to x being an irregular boundary point for the Dirichlet problem in the complementof E.

2.1. Introduction 27

Immediately after, H. Cartan observed, in a letter to Brelot, that an equiva-lent definition of thinness would be to say that E is thin at x if and only if thecomplement of E\x is a neighborhood of x in the weakest topology on C thatmakes all subharmonic functions continuous. Since this topology is strictly finer(in case n > 1) than the Euclidean one, Cartan called it just the fine topology.

It is fairly easy to see that the fine topology is Hausdorff, completely regularand Baire. However, as we will see in Example 2.2.11, the open sets in thistopology can be rather nasty and awkward. In particular they need not have anyEuclidean interior points. Moreover the fine topology has no countable base (hencenot metrizable) and it has no infinite compact sets.

These latter “shortcomings” have discouraged (for a while) any further interestin this topology. And for quite some time, the fine topology was only regarded asa mean of expressing results more elegantly. For instance, an irregular boundarypoint of a domain Ω is now simply a finely isolated point of the complementE = C\Ω. However, in 1954 J.L. Doob [34] discovered a spectacular link betweenthe fine topology and probability theory. He proved that a Borel set Ω is openin the fine topology if and only if a Brownian particle, starting at a point of Ω,remains in Ω with probability 1 through some positive interval of time. About adecade later, Doob [35] showed that the fine topology is quasi-lindelof in a sensemade precise in Theorem 2.2.20.

In a question raised by Ch. Berg, B. Fuglede [55] proved in 1969 that the finetopology is locally connected and enjoys other interesting connectedness proper-ties, cf. Section 2.2.3. This was the starting point for Fuglede for developing finepotential theory which is exposed in his book from 1972, cf. [56]. Thus despite theabsence of non-trivial finely compact sets and the failure of the countability ax-ioms, it turned out that a major part of classical potential theory can be extendedto fine domains. We will provide an amount of this theory in Section 2.3.

Shortly after that fine potential theory was established, many authors struggledto develop a kind of complex analysis on fine domains in C. The first definition of a”holomorphic” function on a fine domain U ⊆ C was given by Fuglede [60] in 1974,cf. Definition 2.4.1. A second definition was proposed in the same year by Debiardand Gaveau [29]; they called a function finely holomorphic on a finely open setU if it is finely harmonic and satisfies the Cauchy-Riemann equation ∂f = 0 a.ein U in the sense of stochastic differentiation along Brownian paths. Fuglede’sdefinition seemed to be less manageable and had the drawback that it did notallow for a simple proof that the algebra of these functions is closed under uniformlimits. Thus, Lyons [97, 98] and Nguyen-Xuan-Loc [106, 107] adopted the seconddefinition. Using probabilistic methods combined with the theory of uniform al-gebra, Lyons obtained several interesting results. And, few years later, togetherwith A. G. O’Farrell, Lyons discovered the missing link with Fuglede’s approachby proving that the two definitions are equivalent. Fuglede resumed the study offine holomorphy in [62, 61] using analytic methods instead of probability. His the-ory is based on the theory of quasi-continuous Beppo-Levi functions [32], and theCauchy-Pompeiu transform of square integrable functions, relying ultimately on akey lemma of T. Lyons, cf. [61, Lemma 6]. We will summarize the properties of


finely holomorphic functions in Section 2.4 and discuss their connection to Borel’smonogenic functions.

2.2 The Fine Topology

As mentioned in the introduction the fine topology was introduced by H. Cartanin 1940 as the coarsest topology on C which makes every subharmonic functionon C continuous. In this section we sum up some basic properties of this topologydescribing the ground on which fine potential theory is built. [2, 36, 76, 55, 58, 67]

Throughout this chapter (and Chapter 6), all topological notions referring tothe fine topology will be qualified by the term ”fine(ly)” to distinguish them fromthose pertaining to the usual (Euclidean) topology on C. For example, finely openmeans open in the fine topology. Note however that for the pluri-fine topology inCn in chapters 3, 4 and 5 the prefix ”F−” is used instead of ”fine(ly)”.

For any set A ⊂ C, let us use A to designate the complement of A. We denoteby A

′

, A and ∂fA the fine interior, the fine closure and the fine boundary of Arespectively . The set of finely isolated points of A is denoted by i(A), and

b(A) = A\i(A)

designates the finely derived set of A. Following Brelot’s terminology, we call b(A)the base of A. Of course, this concept of a base of a set should not be confusedwith the usual topological notion of a base.

2.2.1 Base Elements and Properties

When studying topological properties of a give topological space, a knowledge ofa base is usually necessary. However, a base is not unique. In fact an infinitenumber of bases, even of different ”sizes”, may generate the same topology. Thus,when studying some specific topological problem one base might be more suitablethan another one. In this subsection we discuss different bases and derive someuseful properties of the fine topology.

It is immediate that the fine topology on C has a subbase consisting of allfinely open sets of the form z ∈ C : ϕ(z) > 0 or z ∈ C : ϕ(z) < 0, whereϕ ∈ SH(C). Sets of the last kind, however, are open in the Euclidian topology,in view of the upper semi-continuity of ϕ. By considering the harmonic functionsz 7→ ±<z en z 7→ ±=z we see that the open cubes (hence also usual open sets)are generated by finite intersection of sets of the form z ∈ C : ϕ(z) > c, c ∈ R.This proves the following.

Proposition 2.2.1. Finite intersections of sets of the form

BϕC

= z ∈ C : ϕ(z) > 0,

where ϕ ∈ SH(C), constitute a base of the fine topology on C.

2.2. The Fine Topology 29

Since for any a ∈ C the function z 7→ |z − a| is subharmonic on C, the openballs B(a, r) are finely open for any r > 0. In particular, as may be seen from theabove discussion, the fine topology is finer than the Euclidean one. In fact, weshall later give an example which shows that it is strictly finer.

If Ω is an open subset of C, then the class SH(Ω) is a priori larger than theclass consisting of those subharmonic functions on Ω that are restrictions to Ω offunctions from SH(C). Nevertheless, if ϕ ∈ SH(Ω), then ϕ is finely continuous inΩ. This follows from the fact that for any disk D ⊂⊂ Ω the function ϕ|D can beextended so as to be subharmonic in C, see [76, Lemma 7.13]. We have therefore

Proposition 2.2.2. Let Ω ⊆ C. Then the restriction to Ω of the fine topologycoincides with the coarsest topology on Ω in which every subharmonic function inΩ is continuous.

Using the above mentioned lemma from [76] one can describe a base of the finetopology as follows.

Theorem 2.2.3. Finite intersections of sets of the form

BϕB(a,r) = z ∈ B(a, r) : ϕ(z) > 0,

where B(a, r) = z ∈ C : |z − a| < r, ϕ ∈ SH(B(a, r)), constitute a base of thefine topology on C.

A detailed proof of this theorem is given in [80, page 178 ]. However, we shallgive in the next chapter (Lemma 3.3.1) an easy proof to the fact that sets of theform Bϕ

B(a,r) constitute even a base of the fine topology.

As a corollary to the above theorem we have the following useful descriptionof a local base.

Corollary 2.2.4. Let ζ ∈ C. Then a fine neighborhood base of ζ is given by setsof the form

n⋂

j=1

z ∈ B : ϕj(z) ≥ −1, (2.2.1)

where B is a ball containing ζ and ϕj ∈ SH(B) with ϕj(ζ) = 0.

Notice that the sets in (2.2.1) are compact in the usual topology. In otherwords, the fine topology has a neighborhood base consisting of Euclidean compactset. This base will turn out to be extremely useful in many situations. For instance,relying on this it is an easy exercise to prove the following.

Theorem 2.2.5. The fine topology in C is Baire; that is, if Un, n ∈ N is acountable collection of finely open finely dense sets in C, then the set

⋂

n Un isdense in C.

Another interesting base of the fine topology which is less encountered in theliterature is given by the following theorem, cf. [96, Theorem 2.3].


Theorem 2.2.6. The following collection is a base of the fine topology

B = ϕ > ψ : ϕ, ψ ∈ SH(C), with ϕ ≥ ψ. (2.2.2)

Proof. By Proposition 2.2.1, the collection of sets ϕ > 0, ϕ ∈ SH(C) is a subbaseof the fine topology. Clearly, the following (larger) family ϕ > ψ : ϕ, ψ ∈SH(C) is also a subbase. As ϕ > ψ = max(ϕ, ψ) > ψ, the collection B is asubbase of the fine topology. To see that B is a base it suffices to check that it isstable under taking finite intersection. This is settled by the following observation

ϕ1 > ψ1 ∩ ϕ2 > ψ2 = ϕ1 + ϕ2 > max(ϕ1 + ψ2, ϕ2 + ψ1), (2.2.3)

where ϕj and ψj , j = 1, 2, are subharmonic on C.

Remark 2.2.7. It follows from the above theorem that a non-empty finely open setmust contain infinitely many points. Indeed, let ϕ, ψ ∈ SH(C), and let a ∈ ϕ >ψ. Then there is an l ∈ R such that a ∈ ϕ > l ∩ l > ψ ⊂ ϕ > ψ. Since theset l > ψ is open in the usual topology, it suffices to check that ϕ > l containsinfinitely many points near a. This follows from the mean-value inequality forsubharmonic functions, cf. Inequality (1.1.1). See also Remark 2.2.19 below.

Since subharmonic functions separate points, the fine topology is Hausdorff.The following corollary gives a stronger separation property.

Corollary 2.2.8. The fine topology is completely regular; that is, if A ⊂ C is afinely closed set and a ∈ A, then there exists a finely continuous function f fromC to the real line R such that f |A = 0 and f(a) > 0.

Proof. By Theorem 2.2.6, there exists ϕ, ψ ∈ SH(C) such that ϕ ≥ ψ and

a ∈ ϕ > ψ ⊂ A.

The function f(z) := ϕ(z) − ψ(z) satisfies the assertion of the corollary.

A number of properties of the fine topology follow from the following wellknown result, cf. [36, page 58].

Theorem 2.2.9. Let E be a polar subset of C and a ∈ C. Then there exists asubharmonic function ϕ ∈ SH(C) such that ϕ = −∞ on E\a while ϕ(a) 6= −∞.

We should mention at this point that, although still valid for the potentialtheory of Rn, Theorem 2.2.9 has no analog in pluripotential theory. This is be-cause pluripolar sets in Cn, n > 1, often propagate. In fact, understanding thisphenomenon of propagation constitutes the central problem in this thesis.

Corollary 2.2.10. 1) If E ⊂ C is polar, then E is finely closed. Moreover, E hasno fine limit points in C, i.e, E is finely discrete.2) A set is finely compact if and only if it is finite.3) The fine topology is not separable.4) The fine topology is not first countable; that is, no point of C has a countablefundamental system of fine neighborhoods.5) The fine topology is not metrizable.


Proof. 1) Let a ∈ C\E and let ϕ ∈ SH(C) be the function provided by Theorem2.2.9. Then the set ϕ > ϕ(a) − 1 is a fine neighborhood of a that does notintersect E. Hence C\E is finely open.

By the first assertion in 1), the polar set E\z, z ∈ C, is finely closed. Henceno point z in C can be a fine accumulation point of E.

2) Any finite set is finely compact. Conversely, suppose that K is an infinitefinely compact subset of C. Then there is an infinite countable sequence (an)n≥1

in K. The set Vk = C\⋃∞

n6=kan is finely open, because countable sets are polarand hence finely closed in view of 1). Observe now that Vk : k ∈ N forms acover of K which has no finite subcover.

3) Countable sets are polar, hence finely closed by 1).

4) Suppose that Un, n ≥ 1, is a countable fundamental system of fine neighbor-hoods of a point z. In view of Remark 2.2.7, for each n, there exists zn ∈ Un\z.Since countable sets are polar and hence finely closed by 1), the set zn : n ≥ 1is a fine neighborhood of z which can not contain any Un.

5) follows from 4).

Example 2.2.11. 1) Let (an)n≥1 be a sequence of points in the punctured unitdisk D(0, 1)\0 that converges to 0, and form the series

ϕ(z) =∞∑

k=1

log |z − ak|

2k log |ak|, z ∈ C. (2.2.4)

Then ϕ ∈ SH(C), ϕ(ak) = −∞ for every k ≥ 1 and ϕ(0) = 1. This functionis discontinuous at 0. Accordingly, the fine topology is effectively strictly finerthan the Euclidean one. Observe also that the sequence (an)n≥1 has no fine limitpoints, in view of Corollary 2.2.10.2) Let (wn)n≥1 be a countable dense sequence in the closed unit disk D(0, 1), andlet (an)n≥1 be strictly positive numbers such that

∑

n an < ∞. Define ϕ : C −→[−∞,+∞[ by

ϕ(z) =∑

n≥1

an log |z − wn|, z ∈ C. (2.2.5)

Then:a) ϕ is subharmonic in C and ϕ 6≡ −∞.b) ϕ = −∞ on an uncountable dense subset of D(0, 1).c) ϕ is discontinuous almost everywhere on D(0, 1).If c ∈ R, then z ∈ D(0, 1) : ϕ > c is, in view of a) and b), a finely open whichhas no usual interior points. See [117, page 41-42] for the proof of a), b) and c).3) Let E be the sequence Q×Q, where Q is the set of rational numbers. Then E isa polar subset of C and there exists a subharmonic function ϕ ∈ SH(C) (ϕ 6≡ −∞)such that E ⊂ ϕ = −∞. Observe that if c ∈ R, then the set ϕ > c is a finelyopen and again without usual interior points.


2.2.2 Thinness

Definition 2.2.12. A subset E ⊂ C is said to be thin at a point a if either ais not a limit point of E, or else if there exists a subharmonic function ϕ in aneighborhood of a such that

lim supz→a, z∈E\a

ϕ(z) < ϕ(a). (2.2.6)

It was proved by Brelot [17] that in the above definition one may choose thefunction ϕ in such a way that the lim sup in the left hand side of (2.2.6) is −∞while ϕ(a) = 0.

Example 2.2.13. If E ∈ C is a polar set, then E is thin at every point z ∈ C.This follows by application of Theorem 2.2.9. A less obvious and a typical exampleof a thin set, say at the origin, is given after Theorem 2.2.16.

The following properties follow immediately from the definition of thinness.

Proposition 2.2.14. a) The union of a finite number of sets that are thin at agiven point is thin at this point.b) A set E ⊂ C is thin at a point a ∈ E if and only if there exists an open setΩ ⊃ E\a which is thin at a.

A less obvious and very useful result is the next theorem. Its proof is basedon the fact that thinness is invariant under contractions, cf. [76, Theorem 10.14].Another proof of it is implicit in our proof of Lemma 3.4.4.

Theorem 2.2.15. If E ⊂ C is thin at a, then there are arbitrarily small circle∂D(a, r) such that ∂D(a, r) ∩ E = ∅

An immediate consequence of this theorem is that a connected set containingmore than one point in C is not thin at every point of its closure. It should bementioned here that, as provided by the Lebesgue’s example, cf. [76, page 175],this conclusion and Theorem 2.2.15 fail for the case Rn, n ≥ 3. See also Example3.4.13 in the next chapter.

Theorem 2.2.16. (Wiener’s criterion) Let E ⊂ C, and let ζ0 ∈ C . define

En = z ∈ E : 2−(n+1) ≤ |z − ζ0| ≤ 2−n. (2.2.7)

Then E is thin at ζ0 if and only if∑

n≥1

n

log(1/Cap∗(En))<∞. (2.2.8)

Here Cap∗(En) denotes the outer Logarithmic capacity of the set En. If

Cap∗(En) = 0,

then the corresponding term in the sum (2.2.8) is considered to be equal to zero.See [2, page 218].


Example 2.2.17. Let zn = 12 (2−n + 2−(n+1)), n = 1, 2, ..., and rn = 2−n3

. PutE =

⋃

n D(zn, rn), where D(zn, rn) denotes the disk with center zn and radiusrn. Recall that the Logarithmic capacity of D(zn, rn) is equal to rn. Since thefollowing series

∞∑

n=1

n

log(1/rn)(2.2.9)

is convergent, the Wiener’s criterion shows that⋃

n D(zn, rn) is thin at 0.

The following theorem of H. Cartan gives a characterization of thinness interms of the fine topology.

Theorem 2.2.18. A set E ⊂ C is thin at a point a ∈ C if and only if a is not afine limit point of E. Equivalently, C\(E\a) is a fine neighborhood of a.

Notice that the complement E of the set E in example 2.2.17 is, in view ofTheorem 2.2.18, a fine neighborhood of 0 which is closed in the usual topology.See also the observation after Corollary 2.2.4.

Remark 2.2.19. Another interesting topology that should be compared with thefine topology is the so called ordinary density topology (in e.g. R2). The neigh-borhoods of a point a ∈ C in this topology are the sets E (a 6∈ E) such that a isa dispersion point of E with respect to the Lebesgue measure λ of R2; That is,

limr→0

λ∗(E ∩ B(a, r))

λ(B(a, r))= 0, (2.2.10)

where λ∗ is the outer Lebesgue measure. Relying on Wiener’s criterion, Fugledeproved in [55] that if a set E is thin at a (a 6∈ E), then a is a dispersion point ofE. In other words, the ordinary density topology is finer than the fine topology.In particular, every point of a finely open set has Lebesgue density 1.

The continuous functions with respect to the ordinary density topology, la-beled approximately continuous, were introduced and studied by Denjoy [31] inhis famous paper from (1915). Denjoy proved, in particular, that these functionsare of Baire class 1. However, although the approximately continuous functionswere extensively studied immediately after Denjoy’s paper, a systematic study ofthe density topology has started only in the fifties. The interested reader may finda detailed study of these concepts in [96, Chapter 6]. See also [73].

Theorem 2.2.20. (quasi-Lindelof property). An arbitrary union of finely opensubsets of C differs from a countable subunion by at most a polar set.

The following corollary will be useful in the next section

Corollary 2.2.21. Any finely open set can be written as the union of a (Euclidean)Fσ set and a polar set.

Proof. Let U be a finely open subset of C. By Corollary 2.2.4, U =⋃

z∈U K′

z,where Kz is a compact fine neighborhood of z. According to Theorem 2.2.20,there is a countable sequence zn and a polar set P such that U =

⋃

nK′

zn

⋃

P =⋃

nKzn

⋃

P .


Remark 2.2.22. The statement of the above corollary holds for finely closed setsas well. In fact, a fundamental theorem of Choquet asserts that a finely open(finely closed) set differ from some usual open (closed) by a set of arbitrary smallcapacity. We shall omit any further discussion in this direction and we refer to [2].

In view of the above remark one can expect that finely continuous functionsshould not differ much from usual continuous ones. We will discuss this in Section2.3.2.

2.2.3 Connectedness in the Fine Topology

Recall once more that, except when otherwise indicated, all the results in thischapter are valid in Rn, n ≥ 2.

The first serious study of connectedness in the fine topology was carried outby Fuglede [55] around 1969. Answering a question by Ch. Berg, he proved thefollowing.

Theorem 2.2.23. The fine topology on C is locally connected. Moreover, anyusual domain is also a fine domain; that is, finely open and finely connected.

The proof of this theorem is based on deep results from the theory of balayageof measures (to be defined in the next section). An alternative proof which usesthe theory of finely harmonic functions may be found in [56, page 87-92]. In thenext chapter we give yet another proof of Theorem 2.2.23, cf. Corollary 3.4.7,which is completely elementary.

In view of the quasi-Lindelof property, Theorem 2.2.20, the following corollaryis obvious. See also [55].

Corollary 2.2.24. Every finely connected component of a finely open set U ⊂ C

is finely open. Moreover, the set of these components is at most countable.

Perhaps inspired by the result of J.L. Doob [34] mentioned in the introduction,Nguyen-Xuan-Loc and T. Watanabe [105, Theorem 2.4] characterized in 1972 thefine domains as follows.

Theorem 2.2.25. A necessary and sufficient condition for a finely open set U ⊂ C

to be finely connected is that, given any point z ∈ U and any non-empty, finelyopen set V ⊂ U , there is a positive probability that a Brownian particle starting atz should reach the set V before it (possibly) leaves U .

Since Brownian trajectories are continuous (and with probability 1 even Lips-chitzian of order < 1

2 ), it is quite easy, as observed by Fuglede [58, page 263], toprove the following.

Corollary 2.2.26. Every fine domain is arcwise connected in the usual topology.

Remark 2.2.27. In the same paper, Fuglede gave an alternative proof of Corollary2.2.26 without recourse to probability. He made use of Theorem 2.2.15 in anessential way. Even stronger, he proved that any two points in a fine domain

2.3. Finely Subharmonic Functions 35

U ⊂ C can be joined by a finite polygonal path in U . Since Theorem 2.2.15 breaksdown in Rn, n ≥ 3, the polygonal connectedness for a fine domain U ⊂ Rn, n ≥ 3remained open.

Using probabilistic methods, Lyons [97] proved polygonal connectedness andbrought the following precision to Corollary 2.2.26 (also valid for U ⊂ Rn, n ≥ 3).See also [67] for a non-probabilistic proof.

Theorem 2.2.28. Every point of a fine domain U ⊂ C has a fine neighborhoodV ⊂ U such that any two point z, w ∈ V can be joined by a polygonal path in Uconsisting of just two straight segments [z, ζ] and [ζ, w] of equal length, and suchthat the angle between either of them and the segment [z, w] is < θ (θ > 0 beingprescribed).

We end this subsection with the following result which will be reconsidered inthe next chapter (Theorem 4.2.5). See [56, Theorem 12.2] and [55, Theorem 6].

Theorem 2.2.29. Let U ⊂ C be a finely open set and E a polar set. Then U\Eis finely connected if and only if U is finely connected.

2.3 Finely Subharmonic Functions

Throughout this section Ω denotes an open subset of the complex plane with non-polar complement. A finely subharmonic function in a finely open set U ⊂ Ωis loosely speaking a finely upper semi-continuous function which satisfies thesubmean value with respect to certain measure called swept-out measure. To givea precise definition we must start with the concept of reduced and swept-outfunctions leading to this measure. Later on we shall give a characterization offinely subharmonic functions which does not involve the sweeping-out process (cf.Theorem 2.3.20).

2.3.1 Swept-out Measure

Let SH−(Ω) denote the collection of all non-positive subharmonic functions on Ω.

Definition 2.3.1. If u ∈ SH−(Ω) and E ⊆ Ω, then the reduced function (or“reduite”) of u relative to E is defined by

REu (z) = supv(z) : v ∈ SH−(Ω) and v ≤ u on E, z ∈ Ω. (2.3.1)

It follows from Theorem 1.1.5 that the upper semicontinuous regularization REu

is subharmonic on Ω. We call REu the swept-out or balayage of u relative to E in

Ω. It is the biggest non-positive subharmonic function in Ω which minorizes u onE. When u is the Green potential Gµ of a measure µ on Ω, the swept-out function

REu is the Green potential of a certain measure on Ω denoted by µE and called the

swept-out of µ on E (relative to Ω). A detailed study of reduced functions was


originated by Brelot in [15]. We refer the reader to [2, 76, 36] for a good expositionof the theory of balayage.

Thus this operation of sweeping out of subharmonic functions induces in anatural way a sweeping of measures. In the special case when the measure µ isthe unit mass placed at some point z, the following result, fundamental for thedevelopment of fine potential theory, goes back to Brelot [15], see also [2, page275].

Theorem 2.3.2. If E ⊆ Ω, then there exists a unique Radon measure εEz such

that

REu (z) =

∫

u(y)dεEz (y), (z ∈ Ω). (2.3.2)

For all u ∈ SH−(Ω).

The measure εEz is called the swept-out measure of the Dirac mass εz onto E.

The next theorem, due to Brelot [13], summarizes some fundamental properties ofthe swept-out measure εE

z .

Theorem 2.3.3. Let E ⊆ Ω ⊂ C. Thena) εE

z is carried by b(E).b) εE

z = εz if and only if z ∈ b(E).c) εE

z (P ) = 0 for any polar subset P with z 6∈ P .

These two theorems have led to an extensive theory of balayage of measures,generally carried out in the framework of harmonic spaces by R. M. Herve [75]and Boboc-Constantinesco-Cornea [9]. See also [27]. The reader not familiar withaxiomatic potential theory may find an amount of this theory in Doob [36, ChapterX].

The next proposition gives other interesting properties of the finely harmonicmeasure. Assertions 1) and 2) are due to Fuglede. 3) is a reformulation of b) inthe above theorem.

Proposition 2.3.4. Let U ⊆ Ω be finely open.

1) If U is a fine domain, then the measures εΩ\Uz , z ∈ U , have all the same null

sets.2) ε

Ω\Uz is carried by the fine boundary of the finely connected component of U that

contains z.3) ε

Ω\Uz 6= εz if and only if Ω\U is thin at z.

It should be mention at this point that the measure εEz boils down to the

usual harmonic measure of Ω\E when E is closed and z 6∈ E, cf, e.g [36, page157]. Thus, when E is finely closed εE

z may be called the finely harmonic measurefor the finely open set Ω\E relative to Ω. Incidently, a recent result of S. Roy[128] asserts that these measures are exactly the extreme Jensen measures forsubharmonic functions.


2.3.2 Definition and Basic properties

The submean value property with respect to the measure dεEz , z 6∈ E involves

integrals such as∫

E ϕdεEz where E is finely closed. The measure dεE

z is a Borelprobability measure, but not all finely closed sets are Borel sets. However, sincedεE

z puts zero mass on each polar set, cf. Theorem 2.3.3, it is clear that dεEz has

a natural extension to the following σ-algebra

PB = σ − algebra generated by the Borel sets and the polar sets. (2.3.3)

It follows from Corollary 2.2.21 that PB contains the fine Borel sets; that is, theσ-algebra generated by the finely open sets. In other words, fine Borel sets aremeasurable with respect to εE

z . In particular, every finely semicontinuous functionis εE

z -measurable for any E ∈ PB and z 6∈ E.

Definition 2.3.5. A function ϕ : U −→ [−∞,+∞[ defined on a finely open setU is said to be finely hypoharmonic if1) ϕ is finely upper semicontinuous; that is, ϕ < c is finely open for every c ∈ R.2) Those finely open sets V with closure V ⊂ U for which

ϕ(z) ≤

∫

∂f V

ϕdεVz , ∀z ∈ V. (2.3.4)

form a base for the fine topology in U .3) If moreover ϕ 6≡ −∞ on every finely connected component of U , then we call ϕfinely subharmonic.

We shall let FSH(U) designate the set of finely subharmonic functions on U . Itis illuminating to notice that the base in question depends a priori on the functionϕ. Therefore, it is, for instance, by no means obvious that the sum of two finelysubharmonic functions is finely subharmonic. However, it is a deep result thatfor a finely subharmonic function, the submean property 2.3.4 holds for a largercollection of finely open sets than required. Namely we have the following

Theorem 2.3.6. [56, Lemma 9.5] Let ϕ : U −→ [−∞,+∞[ be a finely subhar-monic function on a finely open set U . Then

ϕ(z) ≤

∫

∂f V

ϕdεVz , ∀z ∈ U, (2.3.5)

for every finely open set V with V ⊂ U and such that ϕ is bounded above in V .

About two decades later, Fuglede extended the above theorem (in the planarcase) to include every finely open set V (independent of ϕ) with fine closure Vcontained in U . See [68] for a precise formulation.

As an answer to a problem left open by Fuglede in [56], T. Lyons [99] showed byuse of Choquet theory that an equivalent definition of fine subharmonicity would


be to replace 2) in Definition (2.3.5) by the following apparently weaker property:2’) For each point z in U there is a base B(z) of fine neighborhoods of z in U suchthat

ϕ(z) ≤

∫

∂f V

ϕdεVz , for all V ∈ B(z). (2.3.6)

The interested reader may also refer to [95] for the equivalence between the variousdefinitions of fine subharmonicity.

For any family ϕn of finely subharmonic functions on a finely open set Uwhich is finely locally bounded from above one can select a base B(U) of thefine topology on U such that the inequality (2.3.5) holds for all n and for everyV ∈ B(U). Hence, Theorem 2.3.6 has the following easy consequences:

Theorem 2.3.7. [56] 1) FSH(U) is a convex cone, i.e c1u1 + c2u2 ∈ FSH(U) forany c1, c2 ≥ 0 and any u1, u2 ∈ FSH(U).2) If u1, ..., u2 ∈ FSH(U), then the function max(u1(x), ..., u2(x)) belongs also toFSH(U).3) The limit of a monotonically decreasing sequence of finely subharmonic func-tions on a fine domain U is finely subharmonic or identically −∞.4) If U is a fine domain, then FSH(U)∪−∞ is closed under finely locally uniformconvergence.

Definition 2.3.8. A function ϕ : U −→ R defined on a finely open set U is calledfinely harmonic if ϕ and −ϕ are both finely subharmonic, or equivalently if ϕ isfinely continuous and those finely open sets V with closure V ⊂ U for which

ϕ(z) =

∫

∂f V

ϕdεVz , ∀z ∈ V.

form a base for the fine topology in U .

Using Theorem 2.3.2 one can easily prove that in the case of a usual open setU any sub(harmonic) function is finely sub(harmonic). The next theorem withthe remark that follows establishes the converse.

Theorem 2.3.9. [56, Theorem 9.8] Let U be an open subset of C. Then a functionϕ : U −→ R is subharmonic if and only if ϕ is finely subharmonic and, moreover,locally bounded from above in the Euclidean topology.

Remark 2.3.10. It was proved by Fuglede [57] that one may drop the local bound-edness in the above theorem. He also gave examples which prove that local bound-edness can not be removed in higher dimensions, i.e, open subset of Rn, n > 2.

Theorem 2.3.11. [57, Thoerem 2.3] Let ϕ be a finely subharmonic function ona bounded finely open set U ⊂ C. If

f- lim supz→x,z∈U

ϕ(z) ≤ 0, (2.3.7)

for every x ∈ ∂fU , Then ϕ ≤ 0.


Here and elsewhere f- lim sup denotes the lim sup with respect to the fine topol-ogy. Theorem 2.3.11 does not hold in Rn, n > 2, in this form.

A deep result in fine potential theory, cf. [56, Theorem 9.9] asserts that a finelylocally bounded finely subharmonic function can be represented (finely locally) asthe difference of two usual subharmonic functions. As a consequence of this wehave

Theorem 2.3.12. Every finely subharmonic function is finely continuous.

The following results shows that finely subharmonic functions glue together,cf. [56, Lemma 10.1]. This will be one of the decisive tools in the first part ofChapter 3.

Lemma 2.3.13. Suppose that V ⊆ U ⊆ C are finely open sets, and let ψ (respϕ) be a finely subharmonic function on U (resp V ). Assume that:

f- lim supz→x,z∈V

ϕ(z) ≤ ψ(x) for all x ∈ U ∩ ∂fV .

Then the following function Ψ is finely subharmonic in U :

Ψ(z) =

maxϕ(z), ψ(z) if z ∈ V ,ψ(z) if z ∈ UV .

Recall once more that f- lim sup denotes the lim sup with respect to the finetopology, i.e.

infO

supz∈O

ψ(z),

where O ranges over the set of all fine open sets in V which contain z.

The following important theorem is a consequence of fundamental propertiesof the swept-out measure and the quasi-Lindelof property, cf. Theorem 2.2.20. Itwill be very useful in the study of pluripolar hulls in Chapter 5.

Theorem 2.3.14. [56, page 158]. Let h : U −→ [−∞,+∞[ be a finely subhar-monic function on a finely open set U ⊂ C. Then the set z ∈ U : h(z) = −∞is a polar subset of U .

We end this section by the following removable singularity theorem for finelyharmonic functions.

Theorem 2.3.15. Let ϕ be finely harmonic in U\E, where E denotes a polarsubset of the finely open set U . Then ϕ has a finely harmonic extension to U if(and only if) ϕ is bounded in some fine neighborhood of each point of E. Theextension is then unique and given by

ϕ(a) = f- lim supz→a,z∈U\E

ϕ(z), a ∈ E (2.3.8)


2.3.3 Continuity and Fine Differentiability Properties

In view of the striking result (Theorem 2.3.12) that finely subharmonic functionsare finely continuous, it is tempting to surmise that finely harmonic functionswould at least be finely differentiable. But this is not the case. Thus, while theusual harmonic functions are infinitely differentiable and even real analytic, thefinely harmonic ones are not.

As we mentioned in Remark 2.2.19, if ϕ is finely continuous, then ϕ is ap-proximately continuous and hence of Baire class 1 in view of Denjoy’s result. Inparticular, every finely subharmonic function is, by Theorem 2.3.12, of Baire class1. The next fundamental result establishes that fine continuity is not very far fromusual continuity, cf. [60, Lemma 1].

Theorem 2.3.16. (The Brelot property) Consider a countable family of finelycontinuous functions fn : U −→ C (U finely open in C). Every point of U hasa fine neighborhood K ⊂ U (K a standard compact set, if we like) such that therestriction of each fn to K is continuous in the standard topology.

The following result is proved in [30].

Theorem 2.3.17. Let K ⊂ C be a compact set, and h be a continuous functionon K. Then the following are equivalent1) f is finely harmonic in the fine interior K

′

of K.2) f can be uniformly approximated on K by a sequence of usual harmonic func-tions hn defined in Euclidean neighborhoods Wn of K.

Using a kind of localization result for harmonic approximation, Gauthier andLadouceur proved in [72] that Theorem 2.3.17 is still valid if one assumes that Kis merely a closed subset of Rn, n ≥ 2. The following fine local version of Theorem2.3.17 has been given in [57, Theorem 4.1]

Theorem 2.3.18. A function h defined in a finely open set U ⊆ C is finelyharmonic if and only if every point of U has a compact fine neighborhood K ⊂U such that h|K is the uniform limit of usual harmonic functions hn defined inEuclidean neighborhoods Wn of K.

Theorem 2.3.17 was proved by Debiard and Gaveau using probabilistic meth-ods. In [63], Fuglede devised an alternative proof which also works for the finelysubharmonic case. His proof uses the theory of balayage and the Hahn-Banachtheorem relying, ultimately, on a deep study of what he calls a ”localization oper-ator”.

Theorem 2.3.19. Let K ⊂ C be a compact set, and h be a continuous functionon K. Then the following are equivalent1) f is finely subharmonic in the fine interior K

′

of K.2) f can be uniformly approximated on K by a sequence of usual subharmonicfunctions hn defined in Euclidean neighborhoods Wn of K.

2.4. Finely Holomorphic Functions 41

In view of the the Brelot property (Theorem 2.3.16), the following corollaryto Theorem 2.3.19 is immediate. As we promised at the beginning of Section 2.3,this may serve as an equivalent definition for fine subharmonicity.

Theorem 2.3.20. A function h defined in a finely open set U ⊆ C is finelysubharmonic if and only if every point of U has a compact fine neighborhood K ⊂ Usuch that h|K is the uniform limit of usual subharmonic functions hn defined inEuclidean neighborhoods Wn of K.

Let f be a real valued function on a finely open set U ⊂ C. We say that f isfinely differentiable at a point a ∈ U if there exists a vector ∇f(a) ∈ C (called thefine gradient of f at a) such that

f(z)− f(a) − 〈z − a,∇f(a)〉

|z − a|(2.3.9)

converges to 0 As z converges finely to a. (If z = x + iy, w = u + iv, then〈z, w〉 = xu+ yv). In view of Cartan’s theorem (cf. e.g [76, Theorem 10.15]), thisamounts to saying that a has a fine neighborhood V such that the above expressionconverges to 0 as z converges (in the usual sense) to a in V .

The following theorem was proved in [64].

Theorem 2.3.21. Every finely subharmonic function in a finely open set U isalmost everywhere (w.r.t Lebesgue measure) finely differentiable in U .

A. M. Davie and B. Øksendal [28] constructed an example of a finely harmonicfunction which is not everywhere finely differentiable.

2.4 Finely Holomorphic Functions

As pointed out in the introduction the study of finely holomorphic functions wascarried out along two different lines. The methods used by Debiard and Gaveau[29], and likewise by Lyons [97, 98] involved uniform algebras combined with prob-abilistic tools like stochastic integration and the famous Ito’s formula. We shallavoid this probabilistic approach here and follow the alternative presentation ofFuglede [60, 61, 66]. The later involves analytical methods, notably the Cauchy-Pompeiu transform. It is much more convenient and natural for a complex analyst.Thus our main reference here will be [61]. In fact almost all the results presentedbelow are included in [61].

In the sequel U will always denote a finely open subset of C

The characterization of fine harmonicity in Theorem 2.3.18 by finely local uniformapproximation by harmonic function suggested in [60] the following definition.

Definition 2.4.1. A function f : U −→ C is called finely holomorphic if everypoint of U has a compact (in the usual topology) fine neighborhood K ⊂ U suchthat the restriction f |K belongs to R(K).


Here R(K) denotes the uniform closure of the algebra of all restrictions to Kof rational functions on C with poles off K, or equivalently, in view of Runge’stheorem, of holomorphic functions in open neighborhoods of K in C.

Notice that by Definition 2.4.1 a holomorphic function f in a usual domain Uis of course finely holomorphic. See also Corollary 2.4.6 below for the conversestatement.

To give an idea of how a finely holomorphic function might look like, it isilluminating to start at least with the following simple example which is a slightmodification of an example in [61, page 74].

Example 2.4.2. Let an > 0, n = 1, 2... be a sequence of positive numbers suchthat

∑∞n=1 an < +∞. Let zn = 1

2 (2−n + 2−(n+1)), n = 1, 2, ..., and rn = 2−n3

.

Put K = D(0, 1)\⋃

n D(zn, rn), where D(zn, rn) denotes the disk about zn withradius rn. Recall that the Logarithmic Capacity of D(zn, rn) is equal to rn. Sincethe following series

∞∑

n=1

n

log(1/rn)(2.4.1)

is convergent, Wiener’s criterion shows that⋃

n D(zn, rn) is thin at 0 ∈ K. ThusK is a (compact) fine neighborhood of 0, cf. Theorem 2.2.18.

Define

f(z) =

∞∑

n=1

an

z − zn. (2.4.2)

Clearly, this series converges absolutely, and locally uniformly, in the open setΩ = C\(znn∈N ∪ 0). Accordingly, f is holomorphic in Ω and meromorphic inC\0. Suppose that an has been chosen in such a way that

∞∑

n=1

an

rn< +∞. (2.4.3)

Then the series fk(z) =∑k

n=1an

z−znof rational functions converges uniformly on

K to f . This shows that f is finely holomorphic in the fine interior K′

of K. Thusaltogether f is finely holomorphic in Ω ∪ 0(= C\znn).

The next theorem shows that finely holomorphic function are closely relatedto finely harmonic function.

Theorem 2.4.3. A function f : U −→ C is finely holomorphic if and only if fand z 7→ zf(z) are both finely harmonic in U .

The theory of finely harmonic functions from the previous section combinedwith Theorem 2.4.3 allows us therefore to derive a number of important propertiesfor finely holomorphic functions.

Corollary 2.4.4. The uniform limit of finely holomorphic functions in a finelyopen set U ⊆ C is finely holomorphic.

2.4. Finely Holomorphic Functions 43

Proof. Follows from Theorem 2.3.7 4).

Corollary 2.4.5. Let E be a polar set contained in U . If f : U −→ C is finelylocally bounded in U and finely holomorphic in U\E, then f extends to a uniquefinely holomorphic functions in U .

Proof. Follows from Theorem 2.3.15.

Corollary 2.4.6. Let U ⊆ C be a usual open set. Then a function f : U −→ C

is finely holomorphic in U if and only if it is holomorphic.

Proof. Follows from the corresponding version of Theorem 2.3.9 for finely harmonicfunctions.

The following theorem is the fine analog of the classical open mapping theoremfor holomorphic functions. It was first proved by Debiard and Gaveau [29]. Seealso [60].

Theorem 2.4.7. Let f : U −→ C be a non-constant finely holomorphic function.Then the image f(V ) of every finely open set V ⊂ U is finely open. Moreoverf−1(E) is a polar subset of U for every polar set E ⊂ C.

Theorem 2.4.8. A function f : U −→ C is finely holomorphic if and only ifevery point of U has a fine neighborhood V ⊆ U in which f coincides with theCauchy-Pompeiu transform of some function ϕ ∈ L2(C) with ϕ = 0 a.e. on V

f(z) =

∫

C

1

z − ζϕ(ζ)dλ(ζ), z ∈ V.

Theorem 2.4.8 is the heart of Fuglede’s theory. it allowed him, in particular,to recover the result of Lyons [98] that every finely holomorphic function f in Uhas a fine derivative f ′ at every point of U and f ′ is finely holomorphic in U ; thatis, at every point z0 ∈ U the fine limit

f ′(z0) = f- limz→z0

f(z) − f(z0)

z − z0(2.4.4)

exists and is finely holomorphic in U . This should be interpreted as follows: Toany ε > 0 there shall correspond a fine neighborhood V ⊂ U of z0 such that

∣

∣

∣

∣

f(z) − f(z0)

z − z0− f ′(z0)

∣

∣

∣

∣

< ε for all z ∈ V. (2.4.5)

This is further equivalent, in view of Cartan’s theorem (cf. e.g [76, Theorem10.15]), to saying that z0 has a compact fine neighborhood K such that

f ′(z0) = limz→z0,z∈K

f(z) − f(z0)

z − z0, (2.4.6)


where the limit here is taken in the usual sense. Thus a finely holomorphic func-tion is infinitely finely differentiable, and all its fine derivatives f (n) are finelyholomorphic.

In fact finely holomorphic functions can be characterized as follows, cf. [62].

Theorem 2.4.9. A function f : U −→ C is finely holomorphic if and only if fis a fine C1-function in the complex variable sense. Explicitly

f ′(z) = f- limw→z

f(w) − f(z)

w − z(2.4.7)

exists for every z ∈ U and f ′ is finely continuous in U .

Theorem 2.4.9 may thus serve as an alternative definition of fine holomorphy.Observe also that taking this as a primary definition is perhaps more natural sinceit is similar to the classical Cauchy’s definition of usual holomorphic functions.

The next corollary follows by application of the chain rule of fine differentiation.See also [57, page 126].

Corollary 2.4.10. Let f : U −→ C and f : V −→ C be finely holomorphic withf(U) ⊂ V . Then g f is finely holomorphic in U , and (g f)′ = (g′)f ′.

Another interesting consequence of Theorem 2.4.8 is the following precise finelocal description of finely holomorphic functions.

Theorem 2.4.11. If f : U −→ C is finely holomorphic, then every point of Uhas a compact fine neighborhood V ⊂ U satisfying a), b), and c) below.a) (Approximation by rational functions.) There exists a sequence of rational func-tions fj with poles off V such that, for each integer n ≥ 0, the n,-th derivative

f(n)j converges uniformly on V to the n,-th fine derivative f (n) as j → ∞.

b) (Asymptotic Taylor expansion.) For any m ≥ 0 there is a constant Am (de-pending also on f and V ) such that the inequality

∣

∣

∣

∣

∣

f(w) −m−1∑

k=1

1

k!(w − z)kf (k)(z)

∣

∣

∣

∣

∣

≤ Am|w − z|m (2.4.8)

holds for every z, w ∈ V .c) (C∞-extension.) There exists a usual C∞-function f : C −→ C such that f = fon V , and

∂nf = f (n), ∂f = 0 on V (2.4.9)

for every n ≥ 0.

Here ∂ and ∂ denote as usual

∂ =1

2

(

∂

∂x− i

∂

∂y

)

, ∂ =1

2

(

∂

∂x+ i

∂

∂y

)

. (2.4.10)

2.5. Borel’s Monogenic Functions 45

Remark 2.4.12. Property b) in the above theorem shows that the Taylor series off at a given point z0, although divergent in general, represents f asymptotically ina fine neighborhood of z0. In fact it is a deep result of Fuglede that f is uniquelydetermined by the sequence of its fine derivatives f (n)(z0), n ≥ 0, at any givenpoint z0 ∈ U . Explicitly, we have the following uniqueness theorem.

Theorem 2.4.13. A finely holomorphic function f in a fine domain U is uniquelydetermined by the sequence of its fine derivatives f (n)(z0), n ≥ 0, at any given pointz0 ∈ U ; that is, if f (n)(z0) = 0, for all n ≥ 0, at any given point z0 ∈ U , thenf ≡ 0.

The proof of Theorem 2.4.13 follows from property b) of theorem 2.4.11 and acertain properties of the fine Green function studied in [59]. We note in passingthat the sequence f (n)(z0), n ≥ 0 might exhibit a rather wild behavior; in anexample (similar to Example 2.4.2) communicated by Fuglede to P. Pyrih [115] itis shown that for any prescribed sequence of real numbers cn there exists a finelyholomorphic functions f such that the sequence f (n)(0) grows faster than cn.

Theorem 2.4.13 tells us in particular that the set of zeros of a finely holomorphicfunction f has empty fine interior (unless f ≡ 0). In fact, it is shown in [61,Theorem 15] that this set is at most countable. See also [66, page 292].

The next theorem is the analog of the classical local inversion mapping theorem.

Theorem 2.4.14. Let f : U −→ C be finely holomorphic. If the fine derivativef

′

(z0) 6= 0 for some z0 ∈ U , then f is injective in some fine neighborhood V ⊂ Uof z0. Moreover the inverse g := f−1 of f is finely holomorphic in f(V ).

The ”quasi-analytic” property stated in Theorem 2.4.13 suggest that a finelyholomorphic function on a fine neighborhood of the real line might belong to somequasi-analytic class in the sense of Denjoy and Carleman, cf. [120]. However thisis not always the case, see [115] for an interesting discussion on this topic.

An extension of the above theory of finely holomorphic functions to functionsof several complex variables has been made in [65]. Explicitly, if U1 and U2 aretwo finely open subsets of C, then a rich theory of fine holomorphy in U1 × U2,retaining almost all the above results, exists. However, this theory is not very sat-isfactory since the product fine topology in C2 is not biholomorphically invariantas explained in [65, page 144]. In contrast, the pluri-fine topology is biholomor-phically invariant, and this was one of our motivation to study this topology inChapter 3.

2.5 Borel’s Monogenic Functions

The possibility of extending the classical concept of a holomorphic function to setsthat are not necessarily open while retaining its distinctive property of uniqueness,cf. Theorem 2.4.13, was extensively investigated by Borel in the period 1892-1914.Borel’s initial idea, cf. [10], was to generalize the Weierstrass’s concept of holo-morphic continuation by which can be defined single valued maximal holomorphic


function as the collection of ”functional elements” created by power series develop-ments. In other words, Borel believed that it must be possible to continue certainholomorphic functions beyond their maximal domain of existence. This idea wasalmost a heretic one to Weierstrass’s concept of maximal domain of existence.

In contrast to Borel’s deep intuition, Poincare [112] had already constructedanalytic expressions presenting certain singularities and believed that he had suf-ficient grounds for concluding to the impossibility of extending the theory of holo-morphic functions beyond the bounds fixed be Weierstrass; that is, he believedin the impossibility of a process of holomorphic continuation by which one couldextend a function beyond its maximal domain of existence. In this connection it isinteresting to state the following paragraph from the preface of Borel’s book [12]:

”A la suite des traveaux de Poincare que j’ai rappeles il y a un instant, ce pointde vue paraissait universellement admis; mais tandis que Poincare acceullait avecbienveillance le premier essai dans lequel je montrais que les limites imposees parWeierstrass netaient pas aussi infranchissables qu’on l’avait cru, les disciples fidelesde Weierstrass ne consentaient meme pas a discuter; je me rappellerai toujoursl’etonnement avec lequel je vis M. Mittag-Leffler, auquel j’avais essaye d’exposermes projets de recherches, ne faire aucun effort pour entrer dans ma pensee et secontenter de retirer de sa malle un Memoire de Weierstrass pour me montrer unephrase qui devait clore definitivement toute discussion: Magister dixit.”

We shall illustrate Borel’s idea to generalize holomorphic continuation by anexample due to Borel himself and bring some precisions using the the theory offinely holomorphic functions.

Example 2.5.1. Let aj , j = 1, 2... be a dense sequence on the unit circle |z| = 1,and let cj , j = 1, 2... be a sequence of positive numbers such that

∑

j=1

cj <∞. (2.5.1)

Consider the following series

f(z) =∑

j=1

cjz − aj

. (2.5.2)

Clearly, this series converges absolutely, and locally uniformly, in the open setC\|z| = 1. Accordingly, the series f represents a function which is holomorphicboth inside and outside the unit circle and we call these sums f1(z) and f2(z)respectively to distinguish the two. It follows by a result of Goursat, cf. [118,page 114], that each of the two function has radial limit ∞ at each of the pointsan. In particular the unit circle is a natural boundary for both f1(z) and f2(z) inthe sense of Weierstrass; That is f1(z), f2(z) cannot be continued holomorphicallyone into the other. This is an apparent ”paradox” since one can then argue thatf1(z) and f2(z) do not represent the ”same” function even though they are bothrepresentable by the series in (2.5.2).

Borel, in his thesis, was able to show that, under more stringent conditionson the sequence (cj), any two points z, w respectively inside and outside the

2.5. Borel’s Monogenic Functions 47

unit circle, can be joined by an arc of a circle C which will necessarily intersectthe unit circle but on which the series (2.5.2) and the series obtained by termby term differentiation any number of times will converge uniformly. Thus theseries represents a C∞ function at all points of C but is not holomorphic on C,in particular at the point of intersection of C with the unit circle. Moreover, andmost importantly, the function f(z) in (2.5.2) has the property that a knowledgeof f(z) for z on an arbitrarily small open subset of the complex plane determinesf(z) uniquely on the part of the plane where it is defined. In this sense f1(z),f2(z) can thus be considered ”continuation” of each other.

We will now proceed as in example 7.2.2 from chapter 7. Define subharmonicfunctions gj(z) = log |z − aj | − 2j and un by

un(z) =

∞∑

j=n

j−3gj(z). (2.5.3)

The terms in the sum of (2.5.3) are subharmonic and they are negative for |z| < kas soon as j > k. Hence un represents a subharmonic functions in C. Observethat un 6≡ −∞ since un(0) is finite. Let D = (∪nun > −10) \ a1, a2, . . .. Weclaim that D = u1 > −∞. Indeed, observe first that

uk1 = −∞\a1, a2, . . . = uk2 = −∞\a1, a2, . . . (2.5.4)

for any natural numbers k1 and k2. Next, if z0 ∈ ∪nun > −∞ \ a1, a2, . . ..Then there exists, by (2.5.4), a natural number k such that |z0| < k and uk > −∞.Since, as mentioned before, all the terms of the series uk(z0) are negative, a suitabletail, say uN(z0), will be very close to 0. In other words, z0 ∈ uN > −10. Hencez0 ∈ D and consequently D = ∪nun > −∞ \ a1, a2, . . .. Therefore,

C\D = ∩∞n=1un = −∞ ∪ a1, a2, . . .. (2.5.5)

Again, by (2.5.4), we conclude that

C\D = u1 = −∞ ∪ a1, a2, . . . = u1 = −∞. (2.5.6)

This proves the claim. In particular, D is, by Theorem 2.2.29, a fine domain.For every j > 3 there exists 0 < cj < 1 such that if |z − aj | < cj , then

un(z) < −11, for every n ≤ j (2.5.7)

Indeed,∑

k>j k−3gk(z) < 0, while

j−1∑

k=n

k−3gk(z) < log j

j−1∑

k=n

k−3 < 10 log j.

So it suffices to take cj = j−21j3.


Next put cj = 2−j cj and define a function on D by

f(z) =

∞∑

j=1

cjz − aj

, (2.5.8)

We claim that the function f is finely holomorphic on D. Indeed, let z0 ∈ D.Then z0 belongs to the finely open set um > −10 for some m. By inequality(2.5.7), um > −10 ⊂ C \∪j≥mB(aj , cj). Therefore, the compact set K = |z| ≤2|z0| \ ∪j≥mB(aj , cj) is a fine neighborhood of z0. Since the series of f in (2.5.8)is uniformly convergent on the compact set K, the claim is proved.

Note that since D is a polar set (containing the sequence aj), the set F =D ∩ |z| = 1 has length 2π, and f is finely holomorphic at each point of F .Moreover, and most importantly, if a ∈ F then the series f in (2.5.8) is uniquelydetermined by the sequence of its fine derivatives f (n)(a), n ≥ 0, cf. Theorem2.4.13. Furthermore, the above two functions f1(z) and f2(z) are finely holomor-phic continuation of each other. And f1(z), f2(z) should thus be conceived asrestrictions of one and the same function. So, Weierstrass theory of holomorphiccontinuation does not really give a complete picture.

During about two decades, Borel struggled to give his ideas a more general formand a solid foundation. This resulted in the creation of the theory of monogenicfunctions defined on a class of sets broader than open sets, and still possess anumber of important properties usually associated with holomorphic functions.These sets where called Cauchy domains and they are a countable increasingunions of certain Swiss cheeses without interior points (w.r.t the usual topology).We shall not describe the details of Borel’s construction. The interested readeris referred to [11, 12, 137]. A discussion about the connection between Borel’smonogenic functions and finely holomorphic ones can be found in [66, 102].

Chapter 3

The Pluri-fine Topology

In this chapter we study the connectedness properties of the pluri-fine topology.The contents is based on the two papers [48, 49] that were jointly written withJan Wiegerinck.

3.1 Introduction

The pluri-fine topology on an open set Ω in Cn is the coarsest topology on Ωmaking all the plurisubharmonic functions on Ω continuous. Almost all the resultsconcerning the classical fine topology, discussed in Chapter 2 remain valid, andeven with the same proofs. For example, Theorem 2.2.5 and Corollary 2.2.8 clearlyextend to the plurisubharmonic case. In other words, the pluri-fine topology isBaire and completely regular. It was observed by Bedford and Taylor [5], that ithas the quasi-Lindelof property in the sense of Theorem 3.2.5.

In view of the interesting connection between pluripolar hulls and finely holo-morphic functions, cf. [41, 44], and the increasing range of application of finepotential theory, see e.g. [58, 101], it seems quite natural to try to extend these“fine” theories to Cn, n > 1. In fact the paper of Fuglede [65] already containsan attempt to introduce fine holomorphy in Cn. Fuglede compares three possiblefine topologies on Cn: the fine topology on R2n, the pluri-fine topology, and then-fold product topology induced by the fine topology on C. He makes it clearthat the pluri-fine topology is the right one to use. Then he notes that local con-nectedness needs to be established before fine holomorphy, or “fine pluripotentialtheory”, can be developed at all. Problems of pluri-fine-topological nature are alsoeasily encountered in complex analysis and other related areas. See for example[6, 54, 109, 122]

It is the purpose of this chapter to show that the pluri-fine topology enjoyspleasant connectedness properties similar to those in Section 2.2.3. The proofgiven by Fuglede in [55] of the local connectedness of the fine topology in Rn doesnot carry over to the plurisubharmonic case. In fact, Fuglede’s proof was strongly

49

50 Chapter 3. The Pluri-fine Topology

based on the theory of balayage of measures, especially the balayage of the unitDirac measure, cf. Theorem 2.3.2. In particular, the strong subadditivity of theharmonic measure was used in an essential way. As far as we know there is noanalog to Theorem 2.3.2 for plurisubharmonic functions. Moreover, the strongsubadditivity of the relative extremal plurisubharmonic function fails to hold inhigher dimensions, as was proved by Thorbiornson [129].

Moreover, unlike the situation in classical potential theory, the notions of thin-ness and pluripolarity are not equivalent, cf. Example 1.4.1. This means thatpluri-thin sets can not be characterized in terms of capacity, which accounts for abig difference between the pluri-fine and fine topology.

Nevertheless, using elementary properties of finely subharmonic functions, thatwere found by Fuglede [56, 57], we give in Section 3.3 a surprisingly simple proofof the local connectedness of the pluri-fine topology, cf. Theorem 3.3.4.

In Section 3.4 we study the structure of open sets in the pluri-fine topologyin terms of slices. The main result here is proposition 3.4.9. Its proof proceedsby “slicing” and using estimates on subharmonic functions, relying ultimately ona classical harmonic measure estimate of A. Beurling and R. Nevanlinna. Theresults obtained in Section 3.4 will be applied to study finely plurisubharmonicfunctions in the next chapter.

3.1.1 Notation

In order to avoid cumbersome expressions like “locally pluri-finely connected sets”,we adopt the following convention: Topological notions referring to the pluri-fine topology will be qualified by the prefix “F” to distinguish them from thosepertaining to the Euclidean topology. For example, F-open, F-domain (it meansF-open and F-connected), F-component,.... In view of the fact that the pluri-finetopology restricted to a complex line coincides with the fine topology on that line,this convention can be used in the one dimensional setting, where we will workwith the fine topology, at the same time.

3.2 Pluri-thin Sets

The first discussion of pluri-thinness appears perhaps in [122]. The definition ofthinness in classical potential theory can be translated to the plurisubharmoniccase.

Definition 3.2.1. A set E ⊂ Cn is pluri-thin at a point a ∈ Cn if and only ifeither a is not a limit point of E or there exist r > 0 and a plurisubharmonicfunction ϕ on B(0, r) such that


ϕ(z) < ϕ(a). (3.2.1)

The fundamental result of H. Cartan, cf. Theorem 2.2.18, connecting thenotion of thinness and the fine topology extends to the plurisubharmonic case.

3.3. Local Connectedness 51

Theorem 3.2.2. The F-neighborhoods of a point a ∈ Cn are precisely the sets ofthe form Cn\E, where E is pluri-thin at a and a 6∈ E.

The following result (see e.g. [80]) and its corollary assert, that being an F-open set is a local property.

Theorem 3.2.3. Finite intersections of sets of the form

BϕΩ = z ∈ Ω : ϕ(z) > 0,

where Ω ⊆ Cn is open, and ϕ ∈ PSH(Ω), constitute a base of the pluri-finetopology on Cn.

Corollary 3.2.4. If Ω1 ⊆ Ω2 ⊆ Cn are open subsets, then the pluri-fine topologyon Ω1 is the same as the topology on Ω1 induced by the pluri-fine topology on Ω2.

For later reference we recall the following result, cf. [5], which is analogue toTheorem 2.2.20.

Theorem 3.2.5. (Quasi-Lindelof property) An arbitrary union of F-open subsetsof Cn differs from a suitable countable subunion by at most a pluripolar set.

3.3 Local Connectedness

The main result in this section is Theorem 3.3.4, which gives a positive answer toa problem raised by Fuglede in [65]. See also [6, page 62]. Its proof relies on thetwo lemmas below.

Lemma 3.3.1. Sets of the form

BϕΩ = z ∈ Ω : ϕ(z) > 0,

where Ω ⊆ Cn is open, and ϕ ∈ PSH(Ω), constitute a base of the pluri-fine topologyon Cn.

Lemma 3.3.1 was stated by Bedford and Taylor in [5, Theorem 2.3] withoutproof. Since we could not find one in the literature, we prove it here.

Proof. By Theorem 3.2.3, BϕΩ is an F-open set. Let U ⊆ Cn be an F-open set,

and let a ∈ U . By Theorem 3.2.2, the complement E of U is pluri-thin at a. Wewill prove that U contains an F-open neighborhood of a of the form stated inthe lemma. This is trivial if a belongs to the Euclidean interior of U . Considerthe case when a is an accumulation point of E. There exist then δ > 0 and aplurisubharmonic function ϕ on B(a, δ) such that


ϕ(z) < ϕ(a).

Without loss of generality we may suppose that ϕ(E ∩B(a, δ)) ≤ 0 < ϕ(a). SinceCn\(E ∩ B(a, δ)) = U ∩ B(a, δ) ∪ Cn\B(a, δ), we get z ∈ B(a, δ) : ϕ(z) > 0 ⊂U ∩ B(a, δ) ⊂ U . This proves the lemma.


Denote by B = B(0, 1) the open unit ball in Cn, and let ϕ ∈ PSH(B(0, 1))such that 0 ≤ ϕ ≤ 1 on B(0, 1).

The next lemma is the second ingredient in the proof of Theorem 3.3.4.

Lemma 3.3.2. Let U = ϕ > 0 ∩ B(0, 1). If U = V ∪ W , where V and Ware non empty F-open sets such that V ∩W = ∅, then the following function isplurisubharmonic:

ϕV (z) =

ϕ(z) if z ∈ B\W,0 if z ∈ W.

Proof. Let L be a complex line passing through B. We will prove that the function

ϕV,L(z) =

ϕ(z) if z ∈ L ∩ B\WL,0 if z ∈WL,

is subharmonic on L ∩ B. Here ϕ denotes the restriction of ϕ to L ∩ B, andWL := W ∩ L . The set WL is an F-open subset of L ∩ B which may be empty.

Note that ϕV,L is the restriction of ϕV to L ∩ B. Denote by ∂fWL the F-boundary of WL relative to L ∩ B. We claim that ϕ = 0 on ∂fWL. To provethe claim observe first that ∂fWL ⊂ ∂fϕ > 0 and ∂fϕ > 0 = ∂fϕ = 0.Moreover, the set ϕ = 0 is F-closed, which means that ∂fϕ = 0 is a subset ofϕ = 0, and hence ∂fWL ⊂ ϕ = 0.

Next, we can assume that L ∩ B\WL is nonempty, for otherwise ϕV,L ≡ 0hence subharmonic. Using the claim and the fact that ϕ is a non-negative upper-semicontinuous function, we get the following:

lim supz→a,z∈B∩L\WL

ϕ(z) ≤ ϕ(a) = 0, ∀a ∈ ∂fWL,

and clearly,

f- lim supz→a,z∈B∩L\WL

ϕ(z) ≤ 0, ∀a ∈ ∂fWL,

because the ordinary lim sup majorizes the f- lim sup.In view of the claim, the definition of ϕV,L does not change if we replace WL by

its fine closure WL. Since ∂f (L∩B\WL)∩B = ∂fWL, Lemma 2.3.13 applies andϕV,L is therefore finely subharmonic. It is clearly bounded, and hence subharmonicby Theorem 2.3.9.

It is a well known result that a bounded function, which is subharmonic oneach complex line where it is defined, is plurisubharmonic. (see e.g. [85]).

Remark 3.3.3. The proof of Lemma 3.3.2 uses Lemma 2.3.13 and Theorem 2.3.9.These are rather deep results in fine potential theory. However, we do not usetheir full strength. Lemma 3.4.12 below, which has a short direct proof, could beused instead.

Theorem 3.3.4. The pluri-fine topology on an open set Ω in Cn is locally con-nected.

3.3. Local Connectedness 53

Proof of Theorem 3.3.4. Let z0 ∈ Cn and let D be an F-open neighborhood of z0.By Lemma 3.3.1 there exist an open set Ω in Cn and a plurisubharmonic functionϕ ∈ PSH(Ω), such that the set z ∈ Ω : ϕ(z) > 0 is an F-open neighborhoodof z0 contained in D. In view of Corollary 3.2.4 and the fact that the pluri-finetopology is biholomorphically invariant, there is no loss of generality if we assumethat z0 = 0, Ω is the unit ball B(0, 1), and that 0 ≤ ϕ ≤ 1 on B(0, 1). To provethe Theorem we will find an F-open neighborhood of 0 which is F-connected andcontained in Bϕ

B(0,1) := z ∈ B(0, 1) : ϕ(z) > 0.

Denote by P the set of all F-open sets V which contain 0 and for which thereexists a non empty F-open set WV such that Bϕ

B(0,1) = V ∪WV and V ∩WV = ∅.

It follows from Lemma 3.3.2 that the function ϕV is plurisubharmonic for allV ∈ P . Moreover, the family (ϕV )V ∈P is left directed and lower bounded by 0.It is a classical result, see e.g. [76, Theorem 4.15], that the infimum ψ of sucha family exists and is plurisubharmonic on B(0, 1). Now we claim that the setU = z ∈ Bϕ

B(0,1) : ψ(z) > 0 is F-open and F-connected neighborhood of 0.

To prove the claim observe first that for every V , ψ(0) = ϕV (0) = ϕ(0) > 0,which means that U is a non-empty F- open neighborhood of 0. Next, from thedefinition of ϕV in Lemma 3.3.2 we see that ϕV (z) = ϕ(z) > 0 on V and ϕV (z) = 0on B(0, 1)\V . Consequently, U =

⋂

V ∈P

ϕV > 0 =⋂

V ∈P

V and BϕB(0,1) =

⋂

V ∈P

V ∪⋃

V ∈P

WV , where⋂

V ∈P

V ∩⋃

V ∈P

WV = ∅. Therefore U is an element of P . It is

minimal in the sense that it can not be split into two disjoint non-empty F- opensets, which proves the claim and the theorem.

Corollary 3.3.5. Every F-connected component of a F-open set Ω ⊂ Cn is F-open. Moreover the set of these components is at most countable.

The proof of this corollary is the same as the proof given by Fuglede in theclassical case. It uses the quasi-Lindelof property. See Fuglede [55].

Let us mention the following theorem, which is due to Fuglede in the classicalcase, see Theorem 2.2.23.

Theorem 3.3.6. A Euclidean open set U ⊆ Cn is F-connected if and only if Uis connected in the Euclidean topology.

Thanks to the fact that the fine topology on R2n is finer than the pluri-finetopology on Cn, Theorem 3.3.6 is an immediate consequence of Theorem 2.2.23(with C replaced by R2n).

Remark 3.3.7. In Subsection 3.4.3 we will give an alternative, more constructive,proof of Theorem 3.3.4. However, it is important to observe that the proof pre-sented above provides an explicit neighborhood basis of F-connected sets: Eachpoint z in an F-open set Ω has a neighborhood basis consisting of F-domains ofthe form Bϕ

Ω


3.4 Further Results on Connectedness

In this section we will investigate connectedness in terms of slices. The main resultis proposition 3.4.9, which leads to an alternative proof of the local connectednessof the pluri-fine topology. The results in Subsection 3.4.2 will be vital for the theoryof finely plurisubharmonic functions, which is developed in the next chapter.

3.4.1 Harmonic Measure

We fix the following notations: D(a, r) = z ∈ C : |z − a| < r, D = D(0, 1),C(a, r) = |z − a| = r, while B(a, r) = |z − a| < r ⊂ Cn. For z ∈ C we denoteby <z (resp. =z) the real (resp. imaginary) part of z. The usual distance from apoint z to a set A will be denoted by d(z, A)

Let Ω be an open set in the complex plane C and let E ⊆ Ω. Subharmonicfunctions on Ω are denoted by SH(Ω), while SH−(Ω) = u ∈ SH(Ω) : u ≤ 0.The harmonic measure (or the relative extremal function) of E (relative to Ω) atz ∈ Ω is defined as follows (see, e.g. [117])

ω(z, E,Ω) = supu(z) : u ∈ SH−(Ω), lim supΩ3v→ζ

u(v) ≤ −1 for ζ ∈ E.

This function need not be subharmonic in Ω, but its upper semi-continuous regu-larization

ω(z, E,Ω)∗ = lim supΩ3v→z

ω(v, E,Ω) (≥ ω(z, E,Ω))

is subharmonic, in view of Theorem 1.1.5. If E is a closed subset of Ω, thenω(., E,Ω) coincides with the Perron solution of the Dirichlet problem in Ω\E withboundary values −1 on ∂E ∩ Ω and 0 on ∂Ω\∂E.

Recall the following result, cf. [17] and [23].

Theorem 3.4.1. Let Ω be a bounded open subset of C. If E ⊂ Ω is a Borelset, then there exists an increasing sequence of compact sets Kj ⊂ E such thatω(z,Kj ,Ω)∗ ↓ ω(z, E,Ω)∗.

Let E ⊂ D. We associate to E its circular projection

E = |z| : z ∈ E.

There is extensive literature on harmonic measure and its behavior under geometrictransformations such as projection, symmetrization, and polarization. We refer to[117] and the survey article [7] where a complete bibliography on this subject isgiven.

Our main tool in what follows is the following classical theorem of A. Beurlingand R. Nevanlinna related to the Carleman-Milloux problem, cf. [8] and [104]. Seealso [7].

Theorem 3.4.2. Let F ⊂ D be compact. Let F be its circular projection. Then

ω(z, F,D) ≤ ω(−|z|, F ,D), for all z ∈ D\F.

3.4. Further Results on Connectedness 55

Let us also recall the following formula for the harmonic measure of an interval[3].

Proposition 3.4.3. Let D(0, R) ⊂ C be a disk with radius R, and let 0 < κ <r < R. Then

ω(0, [κ, r],D(0, R)) = −2

πarctan(

R + κ

R + r

√

r

κ− 1).

3.4.2 Estimates for Subharmonic Functions

Lemma 3.4.4. For every d < c < 0 there exists κ > 0 such that for everyϕ ∈ SH−(D) with ϕ(0) > c and for every point a in the F-open set

V = z ∈ D(0, 1/8) : ϕ(z) > c,

the setΩ = z ∈ D : ϕ(z) ≥ d

contains a circle C(a, δϕ,a) with radius δϕ,a > κ.

Proof. After multiplying ϕ by a constant we can assume that d = −1. Moreover,we may assume that the set E = z ∈ D : ϕ(z) < d is non empty since otherwisethe lemma trivially holds.

Let a ∈ V be fixed. We will first prove the following estimate

ϕ(a) ≤ ω(a,Ea ,D(a, 3/4))∗, (3.4.1)

where Ea = a+ |z − a| : z ∈ E ∩ D(a, 3/4).

Let f be the function f(z) = z+a. Note that the circular projection commuteswith f−1, i.e., f−1(E

a) = (f−1(E ∩ D(a, 3/4))). Hence, to prove (3.4.1) it isenough, in view of the conformal invariance of the harmonic measure, to provethat the estimate (3.4.1) holds for the particular point a = 0, i.e.,

ϕ(0) ≤ ω(0, E,D(0, 3/4))∗. (3.4.2)

By Theorem 3.4.1, there is an increasing sequence of compact subset Kj of E

such that

ω(0,Kj ,D(0, 3/4))∗ = ω(0,Kj,D(0, 3/4)) ↓ ω(0, E,D(0, 3/4))∗. (3.4.3)

The equality in (3.4.3) holds because Kj is compact and 0 /∈ Kj . Let ε > 0. Itfollows from (3.4.3) that there exists a natural number j0 such that

ω(0,Kj0 ,D(0, 3/4)) ≤ ω(0, E,D(0, 3/4))∗ + ε. (3.4.4)

Because E is open, we can find a compact set L ⊂ E such that L = Kj0 . ByTheorem 3.4.2 together with inequality (3.4.4) we get

ω(0, L,D(0, 3/4)) ≤ ω(0,Kj0 ,D(0, 3/4)) ≤ ω(0, E,D(0, 3/4))∗ + ε. (3.4.5)


Since L ⊂ E, and ϕ(z) < −1, for all z ∈ E, inequality (3.4.5) implies the followingestimate

ϕ(0) ≤ ω(0, E,D(0, 3/4))∗ + ε. (3.4.6)

As ε is arbitrary, the estimate (3.4.2), and therefore also (3.4.1), follows.Let now α ∈]0, 1/4[ be a constant such that

I = z ∈ D(a, 3/4) : =z = =a, and <a+ α ≤ <z ≤ 1/2 ⊂ Ea .

Then by (3.4.1)

ϕ(a) ≤ ω(a,Ea ,D(a, 3/4))∗ ≤ ω(a, I,D(a, 3/4)). (3.4.7)

Again by the conformal invariance, (3.4.7) yields

ϕ(a) ≤ ω(0, f−1(I),D(0, 3/4)). (3.4.8)

Since f−1(I) = [α, 1/2−<a], it follows that

ϕ(a) ≤ ω(0, [α, 3/8],D(0, 3/4)). (3.4.9)

Let βj ↓ 0 be a sequence decreasing to 0. Since j 7→ ω(0, [βj , 3/8],D(0, 3/4))decreases to −1 (see e.g [76], Theorem 8.38), there exists a constant 0 < κ < 3/8depending only on c but not on the function ϕ such that

ω(0, [κ, 3/8],D(0, 3/4))< c. (3.4.10)

The last inequality, together with (3.4.9), hence shows that for all a ∈ V , theinterval z ∈ D(a, 3/4) : =z = =a, and κ+ <a ≤ <z ≤ 3/8 can not be a subsetof E

a . We conclude that there exists a δϕ,a ∈ [κ, 1/2] such that

z : |z − a| = δϕ,a ⊂ Ω = z ∈ D : ϕ(z) ≥ d. (3.4.11)

For our purposes we don’t need precise estimates for κ, but these can be easilyobtained using the formula of the harmonic measure of an interval, cf. Proposition3.4.3.

Lemma 3.4.5. Every interval in C is F-connected.

Proof. It suffices to prove that the interval [0, 1] is F-connected. Suppose that Eand F are non-empty disjoint F-open subset of [0, 1] with [0, 1] = E ∪ F . Denoteby 1E the characteristic function of E, and observe that it is Lebesgue measurablein view of Corollary 2.2.21. For 0 ≤ x ≤ 1, define f(x) =

∫ x

01Edλ, where dλ is

the Lebesgue measure of the real line. Invoking the Wiener’s criterion, it is aneasy exercise, using the classical Polya’s inequality between Logarithmic capacityand Lebesgue measure, to prove that

f′

(x) = limh→0

1

h

∫ x+h

x

1Edλ = 1E. (3.4.12)


So f is differentiable, and in particular E (and F ) has positive measure. By the

mean value theorem, for every x ∈]0, 1], f(x)x equals 0 or 1. Thus f(x) ≡ x or

f(x) ≡ 0. This is impossible since E and F have both positive measure.

Lemma 3.4.6. Let ϕ ∈ SH(D) such that 0 ≤ ϕ ≤ 1. Let U be the F-open subsetof D where ϕ > 0. Suppose that there exists a piecewise-C1 Jordan curve γ ⊂ D

such that γ ⊂ U . Let Γ be the bounded component of the complement of γ. ThenW = U ∩ Γ is polygonally connected, and hence F-connected.

Proof. We follow Fuglede’s ideas in [58, Section 5]. A square shall be an opensquare with sides parallel to the coordinate axes. The square centered at z withdiameter d will be denoted by Q(z, d), its boundary by S(z, d).

Let n ≥ 1 be a natural number. For every z ∈ γ there exists 0 < dz < 1/nsuch that ϕ > ϕ(z)/2 on S(z, dz) ⊂ D. This may be proved similarly as thecorresponding well-known statement for circles, cf. Theorem 2.2.15. The squaresQ(z, dz) cover γ. By compactness we can select a finite subcover Q(zj , dj), j =1 . . .mn, that is minimal in the sense that no square can be removed withoutloosing the covering property. Now Ωn = ∪mn

j=1Q(zj , dj) is an open neighborhoodof γ, the boundary of which is contained in

ϕ >1

2min

1≤j≤mn

ϕ(zj).

Since γ is locally connected, the boundary of Ωn will consist of two polygonalcurves if n is sufficiently large. One of these components, say, γn is contained inΓ. Denote by Γn the bounded component of the complement of γn.

Let 0 < ε < 12 min1≤j≤mnϕ(zj) and Kn

ε = ϕ ≥ ε ∩ Γn. Then Knε is a

compact subset of D. Since ∂Γn(= γn) is contained in Knε , an easy application

of the maximum principle shows that Knε is connected. Let z1, z2 be points in

Knε . Repeating the above argument, we find for every δ > 0 a polygonal curve C

contained in Knε/2, such that d(z1, C), d(z2, C) < δ. A well known result, cf. [117]

states that for z ∈ D and almost all θ ∈ [0, 2π]

limr→0

ϕ(z + reiθ) = ϕ(z).

The conclusion is that there exists a polygonal line in Knε/2 that connects z1 with

z2. Letting ε → 0 we conclude that U ∩ Γn = ∪ε>0Knε is polygonally connected.

Since every interval is F-connected, cf. Lemma 3.4.5, so is U ∩ Γn. Finally, sinceW =

⋃

n≥1 U ∩ Γn, we conclude that W is F-connected.

As an easy consequence of Lemma 3.4.6 we recover the following known result,cf. Theorem 2.2.23.

Corollary 3.4.7. The fine topology on C is locally connected.

Proof. Let z ∈ C and let U ⊆ C be an F-neighborhood of z. By Lemma 3.3.1 thereexists an F-open F-neighborhood V = Bϕ

Ω ⊆ U of z. Without loss of generalitywe may assume that

V = D(z, 1) ∩ ϕ > 0,


as noted before V contains arbitrarily small circles about z. Let ∂D(z, δ) be oneof them. Then by Lemma 3.4.6, V ∩ D(z, δ) is an F-neighborhood of z which isan F-domain.

Remark 3.4.8. Besides the elementary proof that we presented here there are atleast three proofs of this corollary. The first one was found by Fuglede [55], whogave a second proof in [56], page 92. Fuglede [70] observed, furthermore, that sinceour proof of the local connectedness in Section 3.3 does not use the fact that thefine topology on C = R2 is locally connected, it provides of course (for n = 1) athird proof of that fact.

3.4.3 Structure of F-Open Sets

We start this Subsection with the technical result that was alluded to at thebeginning of Section 3.4.

Proposition 3.4.9. Let U ⊆ Cn be an F-open subset and let a ∈ U . Then thereexist a constant κ = κ(U, a) and an F-neighborhood V ⊂ U of a with the propertythat for any complex line L through v ∈ V the F-component of the F-open setU ∩ L that contains v, contains a circle about v with radius at least κ.

Proof. Let a ∈ U . By Lemma 3.3.1 there is an open set Ω ⊂ Cn, and a plurisub-harmonic function ϕ ∈ PSH(Ω) such that Bϕ

Ω is an F-neighborhood of a containedin U . Replacing ϕ by ϕ − 1, we may assume that Bϕ

Ω = z ∈ Ω : ϕ(z) > −1.Moreover, since the pluri-fine topology is biholomorphically invariant, there is noloss of generality if we assume that a = 0, Ω = B(0, 2), ϕ ≤ 0, and ϕ(0) = −1/4.Let

V = z ∈ B(0, 1/8) : ϕ(z) > −1/2.

Let v ∈ V and let L be a complex line through v. The restriction ϕL of ϕto B(v, 1) ∩ L is subharmonic and satisfies the conditions of Lemma 3.4.4 withc = −1/2. Consequently, there exists a constant κ (not depending on ϕL), suchthat the set z ∈ B(v, 1) : ϕ(z) ≥ −1 ∩ L, and therefore U ∩ L, contains acircle with radius δϕL,v ∈ [κ, 1

2 ] about v. It follows from Lemma 3.4.6 that the setU∩L∩B(v, δϕL,v) is F-connected. This completes the proof of the proposition.

A slightly weaker but easy formulation is as follows. For a point z in an F-open subset Ω ⊂ Cn and L a complex line passing through z, denote by CL theF-component of z in the F-open set Ω ∩ L.

Theorem 3.4.10. Let Ω be an F-open subset of Cn and let z ∈ Ω. Then ∪L3zCL

is an F-neighborhood of z which is F-connected.

Note that CL is F-open in L, because the fine topology is locally connected,in view of Corollary 3.4.7 (or Theorem 2.2.23).


Proof of Theorem 3.4.10. Let V ⊆ Ω be an F-neighborhood of z provided byLemma 3.3.1. Without loss of generality we may assume

V = B(z, 1) ∩ ϕ > 0,

for some ϕ ∈ PSH(B(z, 1)). Recall that for a complex line L through z, CL isthe F-component of Ω ∩ L that contains z. It is immediate that ∪L3zCL is F-connected. We denote by CL the F-component of V ∩ L that contains z. ByLemma 3.4.4 together with Lemma 3.4.6 we can find a constant κ > 0 such thatV ∩B(z, κ)∩L ⊆ CL, for all complex lines L through z. As CL is clearly containedin CL, V ∩ B(z, κ) is a subset of ∪L3zCL. This proves that ∪L3zCL is an F-neighborhood of z.

As an immediate corollary to Theorem 3.4.10 we recover the main result of theSection 3.3:

Corollary 3.4.11. The pluri-fine topology on an open set Ω in Cn is locally con-nected.

The next gluing lemma was used in the proof of Lemma 3.3.2. It is actuallyan easy consequence of Lemma 2.3.13 and Theorem 2.3.9. But, as promised inRemark 3.3.3, we give here a direct proof that avoids heavy use of the fine potentialtheory machinery.

Lemma 3.4.12. Let v ∈ SH(D) for some domain D ⊂ C. Suppose that v ≥ 0and that there exist nonempty, disjoint F-open sets D1, D2 ⊂ D such that

v > 0 = D1 ∪D2.

Then the function v1 defined by

v1(z) =

0 if z ∈ D \D1,

v(z) if z ∈ D1,(3.4.13)

is subharmonic in D.

Proof. For ε > 0 let Di(ε) = Di ∩ v ≥ ε, (i = 1, 2). We claim that D1(ε) isclosed in D. Indeed, take a sequence xn in D1(ε) that converges to y ∈ D. Sincev ≥ ε is closed in D, v(y) ≥ ε. Thus y ∈ D1 ∪D2. Suppose that y ∈ D2. Againthere exists r > 0 such that C(y, r) is contained in the F-open set v > ε/2. ByLemma 3.4.6 the set

U = D(y, r) ∩ v > ε/2

is an F-connected subset of D1 ∪ D2. Since U ∩ D2 is non-empty, U ∩ D1 = ∅.This contradicts the fact that U contains xn for n sufficiently large. Thus y ∈ D1

and hence y ∈ D1(ε), which proves the claim. Similarly, D2(ε) is closed.Now define

vε(z) =

ε if z ∈ D \D1(ε),

v(z) if z ∈ D1(ε).(3.4.14)


The function vε is clearly upper semicontinuous in D and it satisfies the meanvalue inequality in D \D1(ε). Let a ∈ D1(ε) and denote by v(a, r) the mean valueof v over the circle C(a, r). Since D2(ε) is closed, the exists 0 < r0 < d(a, D)such that D(a, r0) ∩ D2(ε) = ∅. Thus for r < r0 we have v ≤ ε on the part ofC(a, r) that lies outside D1(ε). Hence v ≤ vε on C(a, r) for r < r0. Consequently,

vε(a) = v(a) ≤ v(a, r) ≤ vε(a, r).

This proves that vε is subharmonic in D. Finally, the sequence v1/nn decreasesto v1, showing that v1 is subharmonic.

It was proved by Gamelin and Lyons in [71] that an F-open subset of C isF-connected if and only if it is connected with respect to the usual topology onC. The next example shows that this result has no analog in Cn for n > 1.

Example 3.4.13. There exists an F-open set U ⊂ C2, which is connected butnot F-connected. Indeed, consider the set

Γ = (x, y) ∈ C2 : y = e1/x,−1 ≤ x < 0.

As was proved by Wiegerinck [133], one can find a plurisubharmonic functionϕ ∈ PSH(B(0, 2)) such that ϕ|Γ = −∞ and ϕ(0) = 0. Let V = ϕ > −1/2and W = ϕ < −1/2. Let W1 be the connected component of W that containsΓ ∩ B(0, 2), and let V1 the F-component of V that contains 0. Since the pluri-fine topology is locally connected, V1 is F-open. Of course V1 is connected sinceit is already F-connected. Observe now that U = V1 ∪ W1 is F-open and F-disconnected. On the other hand, since W1 ∪0 is clearly connected, the F-openset U = V1 ∪W1 = V1 ∪ (W1 ∪ 0) is connected.

Chapter 4

Finely Plurisubharmonic

Functions

In this chapter we study finely plurisubharmonic functions. The contents is basedon the two papers [49, 50] that were jointly written with Jan Wiegerinck. Wewill keep the notations used in Chapter 3 to distinguish the notions referring tothe pluri-fine topology from those pertaining to the Euclidean one, see Subsection3.1.1.

4.1 Introduction

In the previous chapter we showed that the pluri-fine topology of Cn shares manyproperties with the classical fine topology of Rn, see Subsection 2.2.3. Here we willinvestigate to what extent the results in section 2.3.2 about finely subharmonicfunctions can be extended to the “pluri-fine” case. To do this one has first to choosea “good” definition of finely plurisubharmonic functions. Indeed, the definition ofordinary plurisubharmonic functions by complex lines provides one possibility andcan be easily translated to the pluri-fine setting. But the approximation theoremin Chapter 2 (Theorem 2.3.20), and the third property in Theorem 2.3.7 suggestother choices. However in order to develop a rich theory, the first option, which isof course weaker than the other ones, seems more appropriate.

Since we are motivated by the application to pluripolar hulls, we will focus onthe F-pluripolar sets; that is, the sets where a finely plurisubharmonic functionscan take the value −∞. Using the results obtained in Section 3.4 we show thatF-pluripolar sets have no F-interior points, cf. Theorem 4.2.3. This result has theinteresting consequence that pluripolar sets do not separate F-domains. Theorem4.2.3 appeared in [49] and was already applied to questions about pluripolar hulls.

In Section 4.3 we prove that every bounded finely plurisubharmonic functionscan be F-locally written as differences of two ordinary plurisubharmonic functions.Accordingly, every finely plurisubharmonic functions is pluri-finely continuous (not

61

62 Chapter 4. Finely Plurisubharmonic Functions

just pluri-finely upper semi-continuous by definition). These results together withTheorem 4.2.3 will be used in Section 4.4 to prove that F-pluripolar sets areactually pluripolar in the usual sense. This will have interesting consequences inthe next chapter.

4.2 Finely Plurisubharmonic Functions

As far as we know there is no generally accepted definition of F-plurisubharmonicfunctions. The following, cf. [47], seems quite natural.

Definition 4.2.1. A function f : Ω −→ [−∞,∞[ (Ω is F-open in Cn) is calledF-plurisubharmonic if f is F-upper semicontinuous on Ω and if the restriction off to any complex line L is finely subharmonic or ≡ −∞ on any F-component ofΩ ∩ L.

We shall let F-PSH(U) designate the set of F-plurisubharmonic functions onU.

It follows immediately from this definition, Theorem 2.3.9 and Remark 2.3.10that any usual plurisubharmonic function is F-plurisubharmonic where it is de-fined.

Clearly, an F-plurisubharmonic function f on an F-open set Ω has an F-pluri-subharmonic restriction to every F-open subset of Ω. Conversely, suppose thatf is F-plurisubharmonic in some F-neighborhood of each point of Ω. Then f isF-plurisubharmonic in Ω, see [56, page 70]. We shall refer to this by saying thatthe F-plurisubharmonic functions have the sheaf property.

Using Theorem 2.3.7 one can easily check the following properties. See also[47].

Theorem 4.2.2. 1) F-PSH(U) is a convex cone, i.e c1u1 + c2u2 ∈ F-PSH(U) forany c1, c2 ≥ 0 and any u1, u2 ∈ F-PSH(U).2) If u1, ..., u2 ∈ F-PSH(U), then the function max(u1(x), ..., u2(x)) belongs alsoto F-PSH(U).3) The limit of a monotonically decreasing sequence of F-plurisubharmonic func-tions is F-plurisubharmonic.4) F-PSH(U) is closed under finely locally uniform convergence.

A subset E of Cn is called F-pluripolar if for every point z ∈ E there is anF-open subset U ⊂ Cn containing z and an F-plurisubharmonic function (6≡ −∞)f on U such that E ∩ U ⊂ f = −∞.

The next theorem shows that F-pluripolar have no F-interior points. This willbe used in Section 4.4 to show that these sets are actually pluripolar in the usualsense.

Theorem 4.2.3. Let f be an F-plurisubharmonic function on an F-domain Ω.If f = −∞ on an F-open subset U of Ω, then f ≡ −∞.

4.2. Finely Plurisubharmonic Functions 63

Proof. Without loss of generality we can assume that U is the F-interior of theset f = −∞. Suppose there exists z0 ∈ Ω which is an F- boundary point of U .After scaling we can assume that z0 = 0 and that

V = B(0, 1) ∩ ϕ > 0 ⊂ Ω. (4.2.1)

is an F-neighborhood of 0 defined by a function ϕ ∈ PSH(B(0, 1)) with ϕ(0) = 1,see Lemma 3.3.1. Then

V1/2 = B(0, 1/2) ∩ ϕ > 1/2 (4.2.2)

is a smaller F-neighborhood of 0. Notice that V1/2 ∩U is non empty, because 0 isan F-boundary point of U . For every z ∈ V1/2 ∩ U the function ϕ is defined onB(z, 1/2) and B(z, 1/2)∩ ϕ > 0 is an F-neighborhood of z contained in Ω.

By Lemma 3.4.4 together with Lemma 3.4.6 there exists κ > 0 such that forevery line L passing through z ∈ V1/2 ∩ U there exists δz,L ∈ [κ, 1/2] such thatCz,L = ϕ > 0 ∩B(z, δz,L)∩L is an F-connected F-neighborhood of z in L∩ V .Because z ∈ U , Cz,L meets U in an F-open subset of L. Therefore f ≡ −∞ onCz,L according to Theorem 2.3.14. It follows that f ≡ −∞ on the F-open set

V ∩ B(z, κ) ⊂⋃

L3z

Cz,L.

Now if |z| < κ, then 0 ∈ V ∩ B(z, κ). The conclusion is that 0 ∈ U . This is acontradiction. Hence U = Ω.

Corollary 4.2.4. Let U be an F-domain in Cn, and let E ⊂ f = −∞, wheref is F-plurisubharmonic on U (6≡ −∞). Then U\E is F-connected.

Proof. Suppose that U\E = (V ∪W )\E, where V and W are non-empty F-opensubsets of U such that V ∩W ⊂ E. Define h : U\E → [−∞,∞[ by

h(z) =

0 if z ∈ V \E,

−∞ if z ∈W\E,(4.2.3)

and

f(z) =

f + h if z ∈ V ∪W\E,

−∞ if z ∈ E.(4.2.4)

Then f is F-upper semi-continuous. If we restrict f to a complex line L, it is finelyhypoharmonic. Indeed, on V f is F-plurisubharmonic because V ∩ L is F-openand f = f 6≡ −∞, and in U\V there is nothing to prove because there f = −∞.By Theorem 4.2.3, f ≡ −∞, a contradiction.

Since the restriction of a usual plurisubharmonic function ϕ ∈ PSH(Cn) to anF-domain U ⊂ Cn is F-plurisubharmonic on U , Corollary 4.2.4 has the followingimmediate consequence. See Theorem 2.2.29 for an analogue result.


Theorem 4.2.5. Let U be an F-domain in Cn. If E is a pluripolar set, thenU\E is F-connected.

It should be mentioned that contrary to the situation in the classical fine topol-ogy, the set U\E in Theorem 4.2.5 is not F-open in general. This is of course due tothe fact that the pluripolar set E might not be F-closed as is the case in Example1.4.1.

One more consequence of Theorem 4.2.3 is the following maximum principlefor F-plurisubharmonic functions.

Theorem 4.2.6. Let f ≤ 0 be F-plurisubharmonic function on an F-domain Uin Cn. Then either f < 0 or f ≡ 0.

Proof. Suppose that the F-open set V = z ∈ U : f(z) < 0 is not empty.The function gn = nf is F-plurisubharmonic. Since gn decreases on V , the limitfunction g is F-plurisubharmonic. By Theorem 4.2.3, g ≡ −∞ since it equals −∞in V . Hence f < 0.

4.3 Continuity of Finely PSH Functions

Theorem 4.3.1. Let f be a bounded F-plurisubharmonic function in a boundedF-open subset U of Cn. Every point z ∈ U has then an F-neighborhood O ⊂ Usuch that f is representable in O as the difference between two locally boundedplurisubharmonic functions defined on some usual neighborhood of z.

Proof. We may assume that −1 < f < 0. Let z0 ∈ U , and let V ⊂ U be a compactF-neighborhood of z0. Since the complement V of V is pluri-thin at z0, thereexist 0 < r < 1 and a plurisubharmonic function ϕ on B(z0, r) such that

lim supz→z0,z∈V

ϕ(z) < ϕ(z0). (4.3.1)

Without loss of generality we may suppose that ϕ is negative in B(z0, r) and

ϕ(z) = −1 if z ∈ B(z0, r)\V and ϕ(z0) = −1

2. (4.3.2)

Define

Φ = supψ ∈ PSH−(B(z0, r)) : ψ ≤ −1 on B(z0, r)\V . (4.3.3)

Here PSH−(B(z0, r)) denotes the cone of negative plurisubharmonic functions inB(z0, r).

It is well known that the upper semi-continuous regularization of Φ, i.e. Φ∗(z) =lim supB(z0,r)3v→z Φ(v), is plurisubharmonic in B(z0, r). In view of (4.3.2), we get

ϕ ≤ Φ ≤ Φ∗. In particular − 12 ≤ Φ∗(z0).

As Φ∗(z) = −1 for every z ∈ B(z0, r)\V , we get

f(z) + λΦ∗(z) ≤ −λ for any z ∈ U ∩ B(z0, r)\V and λ > 0. (4.3.4)

4.3. Continuity of Finely PSH Functions 65

Now define a function uλ on B(z0, r) as follows

uλ(z) =

max−λ, f(z) + λΦ∗(z) if z ∈ U ∩B(z0, r),

−λ if z ∈ B(z0, r)\V .(4.3.5)

This definition makes sense because [U ∩ B(z0, r)]⋃

[B(z0, r)\V ] = B(z0, r), andthe two definitions agree on U ∩ B(z0, r)\V in view of (4.3.4).

Clearly, uλ is F-plurisubharmonic in U ∩ B(z0, r) and in B(z0, r)\V , hencein all B(z0, r) in view of the sheaf property, cf. [49]. Since uλ is bounded inB(z0, r), it follows from Theorem 2.3.9 that uλ is subharmonic on each complexline where it is defined. It is a well known result that a bounded function, whichis subharmonic on each complex line where it is defined is a plurisubharmonic, cf.[85]. In other words uλ is plurisubharmonic in B(z0, r).

Since − 12 ≤ Φ∗(z0), the set O = −4 < −1 + 4Φ∗ is an F-neighborhood of

z0. Since Φ∗ = −1 on B(z0, r)\V , it is clear that O ⊂ V ⊂ U .Observe now that −4 ≤ f(z) + 4Φ∗(z), for every z ∈ O. Hence

f(z) = u4(z) − 4Φ∗(z), for every z ∈ O. (4.3.6)

The proof is inspired by [56, page 88-90]. It also shows that f is F-continuousin the F-open set O. In fact, we have the following more precise result.

Corollary 4.3.2. Every F-plurisubharmonic function is F-continuous; that is,continuous with respect to the pluri-fine topology.

Proof. Let f be F-plurisubharmonic in an F-open subset Ω of Cn. Since f isF-upper semicontinuous and f < +∞, the set f < c is F-open for any c ∈ R. Itremains to prove that f > c is F-open. LetM ∈ R, and suppose that there existsz0 ∈ Ω with f(z0) > M . The function g = maxf,M is F-plurisubharmonic andbounded in an F-open neighborhood U of z0. Hence it follows from Theorem4.3.1 that g is F-continuous in a possibly smaller F-neighborhood O ⊂ U of z0.Consequently, the set

z ∈ O : f(z) > M = z ∈ O : g(z) > M, (4.3.7)

containing z0, is F-open and hence an F-neighborhood of z0; and so is thereforef > M.

The following result gives a partial analogue to the Brelot property, cf. Theo-rem 2.3.16. See also [66, page 284] or [60, Lemma 1].

Theorem 4.3.3. (Quasi-Brelot property) Let f be a plurisubharmonic function inthe unit ball B ⊂ Cn. Then there exists a pluripolar set E ⊂ B such that for everyz ∈ B \ E we can find an F-neighborhood Oz ⊂ B of z such that f is continuousin the usual sense in Oz


Proof. Without loss of generality we may assume that f is continuous near theboundary of B. By the quasi-continuity theorem (cf. [80, Theorem 3.5.5], and theremark that follows it) we can select a sequence of relatively compact open subsetωn of B such that the Monge-Ampere capacity C(ωn, B) < 1

n , and f is continuouson B \ ωn. Denote by ωn the F-closure of ωn.

The pluriharmonic measureU∗ωn,B is equal to the pluriharmonic measure U∗

ωn,B ,because for a psh function ϕ the set ϕ ≤ −1 is F-closed, thus ϕ|ωn ≤ −1 ⇒ϕ|ωn ≤ −1. Now, using [80, Proposition 4.7.2]

C(ωn, B) = C∗(ωn, B) =

∫

Ω

(ddcU∗ωn,B)n =

∫

Ω

(ddcU∗ωn,B)n = C∗(ωn, B). (4.3.8)

Let E =⋂

n ωn. By equation (4.3.8), C∗(E,B) ≤ C∗(ωn, B) ≤ 1n , for every n.

Hence E is a pluripolar subset of B.Let z 6∈ E. Then there exists N such that z 6∈ ωN . Clearly, the set B \ ωN is

an F-neighborhood of z. Since f is continuous on B \ωN , it is also continuous onthe smaller set B \ ωN (⊂ B \ ωN).

Remark 4.3.4. Using Theorem 4.3.1, one can easily show that Theorem 4.3.3 ex-tends to the F-plurisubharmonic case.

4.4 F-Pluripolar Sets

In this section we prove that F-pluripolar sets are pluripolar. This will be appliedto the study of pluripolar hulls in the next chapter.

Theorem 4.4.1. Let f : U −→ [−∞,+∞[ be an F-plurisubharmonic function(6≡ −∞) on an F-open and F-connected subset U of Cn. Then the set z ∈ U :f(z) = −∞ is a pluripolar subset of Cn.

Proof. We may assume that f < 0. Let z0 ∈ U such that f(z0) = −∞. Letfn = 1

n max(f,−n). Then −1 ≤ fn < 0. We keep the notations as in the proofof Theorem 4.3.1. In particular, V ⊂ U is an F-neighborhood of z0, and − 1

2 ≤Φ∗(z0). Define a function vn(z) on B(z0, r) as follows.

vn(z) =

max−1, 14fn(z) + Φ∗(z) if z ∈ U ∩ B(z0, r),

−1 if z ∈ B(z0, r)\V .(4.4.1)

Since vn is analogues to the function uλ in (4.3.5), a similar argument showsthat vn ∈ PSH(B(z0, r)). Since fn(z) increases to 0 for every z ∈ U such thatf(z) 6= −∞, it is clear that vn(z) is increasing. Let lim vn = ψ. It is wellknown that the upper semi-continuous regularization ψ∗ of ψ is plurisubharmonicin B(z0, r). Also, the set E = ψ 6= ψ∗ is a pluripolar subset of B(z0, r), byTheorem 4.6.3 in [80].

We claim that ψ∗ = Φ∗ on B(z0, r). Indeed, Observe first that ψ = Φ∗ onB(z0, r) \ f = −∞, because vn = Φ∗ = −1 on B(z0, r)\V . Hence

ψ∗ = Φ∗ on B(z0, r) \ (f = −∞ ∪ E). (4.4.2)

4.4. F-Pluripolar Sets 67

Next, suppose that there exists a ∈ B(z0, r) such that ψ∗(a) 6= Φ∗(a). Clearly,the set ψ∗ 6= Φ∗ is an F-open neighborhood of a, which is contained in f =−∞∪E. But the set f = −∞∪E is obviously F-pluripolar. Hence, it has noF-interior points, in view of Theorem 4.2.3. This contradiction proves the claim.

Fix z ∈ Φ∗ > − 23 ∩ f = −∞. Since Φ∗ = −1 on B(z0, r)\V , it follows

immediately from the definition of vn that ψ(z) = − 14 +Φ∗(z). Hence ψ(z) = − 1

4 +ψ∗(z), according to the above claim. This shows that Φ∗ > − 2

3∩f = −∞ is asubset of E. Since Φ∗ > − 2

3 is an F-neighborhood of z0, we conclude that verypoint z ∈ f = −∞ has an F-neighborhood Oz ⊂ U such that Oz ∩f = −∞ isa pluripolar set. If f(z) 6= −∞ we choose Oz such that Oz ∩ f = −∞ = ∅. Bythe quasi-Lindelof property, cf. Theorem 3.2.5, there is a sequence znn≥1 ⊂ Uand a pluripolar subset P of U such that

U = ∪nOzn ∪ P. (4.4.3)

Hencef = −∞ ⊂ (∪nOzn ∩ f = −∞) ∪ P. (4.4.4)

This completes the proof since a countable union of pluripolar sets is pluripolar.

Remark 4.4.2. Corollary 4.3.2 and Theorem 4.4.1 give affirmative answers to twoquestions in [47].

A weaker formulation of Theorem 4.4.1, but perhaps more useful, is as follows.

Corollary 4.4.3. Let f : U −→ [−∞,+∞[ be a function defined in an F-domainU ⊂ Cn. Suppose that every point z ∈ U has a compact F-neighborhood Kz ⊂U such that f |Kz is the decreasing limit of usual plurisubharmonic functions inEuclidean neighborhoods of Kz. Then either f ≡ −∞ or the set f = −∞ ispluripolar subset of U .

Chapter 5

Finely Holomorphic

Functions and Pluripolar

Hulls

In this chapter we discuss the connection between the theory of finely holomorphicfunctions and the pluripolar hulls. The results presented here are selected fromthe papers [41, 49, 50]. We keep the notations of Subsection 3.1.1.

5.1 Introduction

A subset E ⊂ Cn is said to be pluripolar if for each point a ∈ E there is anopen neighborhood Ω of a and a function ϕ ( 6≡ −∞) plurisubharmonic in Ω,(ϕ ∈ PSH(Ω)) such that

E ∩ Ω ⊂ z ∈ Ω : ϕ(z) = −∞.

It is a fundamental result of Josefson [78] that this local definition is equivalent tothe global one, i.e., in this definition one can assume ϕ to be plurisubharmonic inall of Cn with

E ⊂ z ∈ Cn : ϕ(z) = −∞.

E is called complete pluripolar (in Cn) if for some plurisubharmonic functionϕ ∈ PSH(Cn), we have E = z ∈ Cn : ϕ(z) = −∞. Unlike the situation inclassical potential theory, pluripolar sets often ”propagate”; it may happen thatany PSH function ϕ which is −∞ on a pluripolar set E is automatically −∞ ona larger set. For example, if the −∞ locus of a PSH function ϕ contains a non-polar piece of a complex analytic variety A, then the set z ∈ Cn : ϕ(z) = −∞must contain all the points of A. However, the structure of pluripolar sets maybe much more complicated, cf. [25, 91]. Completeness of pluripolar sets hasreceived growing attention, and in particular cases many results were obtained,

69

70 Chapter 5. Finely Holomorphic Functions and Pluripolar Hulls

see [25, 38, 40, 43, 44, 91, 125, 134, 140]. But our knowledge and understandingof the general situation is fragmentary, and a good characterization of completepluripolar sets is still lacking, even in the case of the graph of an analytic function.

Recently, in [44], Edlund and Joricke have connected the propagation of thegraph of a holomorphic function as a pluripolar set to fine holomorphic continua-tion of the function.

Theorem 5.1.1 (Edlund and Joricke, [44] Theorem 1). Let f be holomorphicin the unit disk D ⊂ C and let p ∈ ∂D. Suppose that f has a finely holomorphiccontinuation F at p to a closed F-neighborhood V of p. Then there exists anotherclosed F-neighborhood V1 ⊂ V of p, such that the graph ΓF (V1) is contained in thepluripolar hull of Γf (D).

The definitions of the pluripolar hull is given in Section 5.2. A detailed discus-sion of the theory of finely holomorphic functions is included in Chapter 2.

In view of this result, it is reasonable to try and investigate the connectionbetween finely holomorphic functions and pluripolar sets. Indeed, using Fuglede’sfundamental work, both in fine potential theory and fine holomorphy, we provedin [41] that graphs of finely holomorphic maps are pluripolar. Accordingly, ageneralized and precise version of Theorem 5.1.1 was obtained. The main argumentused in [41] was a kind of integral representation of finely holomorphic functions, cf.Theorem 2.4.8, together with Theorem 2.3.14. In order to deal with some technicaldifficulties we appealed to the quasi-Lindelof property, cf. Theorem 2.2.20, andthe local connectedness property of the classical fine topology.

Nevertheless, we will leave here the above arguments aside and rely instead onTheorem 4.4.1 from the preceding Chapter. This allows us to recover the mainresults in [41] and even obtain stronger ones. This is done in Sections 5.3, and 5.4.In Section 5.5 we introduce the concept of a finely analytic curve, and relate thisto pluripolar hulls. Finally, we discuss some open problems.

5.2 Preliminaries

5.2.1 Pluripolar Hulls

Let E be pluripolar set in Cn. The pluripolar hull of E relative to an open subsetΩ of Cn is the set

E∗Ω = z ∈ Ω : for all ϕ ∈ PSH(Ω) : ϕ|E = −∞ =⇒ ϕ(z) = −∞.

The notion of pluripolar hull was first introduced by Zeriahi in [139]. The paper[93] of Levenberg and Poletsky contains a more detailed study of this notion.Let f be a holomorphic function in an open set Ω ⊆ Cn. We denote by Γf (Ω) thegraph of f over Ω,

Γf (Ω) = (z, f(z)) : z ∈ Ω.

It is immediate that Γf (Ω) is a pluripolar subset of Cn+1. The pluripolar hull of thegraph of a holomorphic function was studied in e.g. [38, 40, 43, 44, 125, 134, 140].

5.3. Graphs of Finely Holomorphic Functions 71

Of particular interest for our present considerations is the following (see [38,40]).

Theorem 5.2.1 (Edigarian and Wiegerinck). Let D be a domain in C andlet A be a closed polar subset of D. Suppose that f ∈ O(D\A) and that z0 ∈ A.Then the following conditions are equivalent:(1) (z0 × C) ∩ (Γf )∗D×C

6= ∅.(2) the set z ∈ D\A : |f(z)| ≥ R is thin at z0 for some R > 0.

5.2.2 F-Holomorphic Functions

In this subsection we define F-holomorphic functions of several complex variables.The one variable case was treated in Section 2.4

Definition 5.2.2. Let U ⊆ Cn be F-open. A function f : U −→ C is said to beF-holomorphic if every point of U has a compact F-neighborhood K ⊆ U suchthat the restriction f |K belongs to H(K).

Here H(K) denotes the uniform closure on K of the algebra of holomorphicfunctions in a neighborhood of K.

Definition 5.2.3. Let f1 and f2 be F-holomorphic functions on F-domains U1

and U2, respectively, and suppose that the intersection U1 ∩ U2 is non empty andthat f1 = f2 on U1 ∩ U2. Then f2 is called a direct F-holomorphic continuationof f1 to U2, and vice versa.

A fundamental result op Fuglede, cf. [61, Theorem 15], asserts that a finelyholomorphic function of one variable has at most countably many zeros (if 6≡ 0).Accordingly, the direct F-holomorphic continuation is unique in view of Theorem3.4.10.

5.3 Graphs of Finely Holomorphic Functions

The main result in this section is Theorem 5.3.1. It generalizes an earlier resultwhich was obtained in collaboration with Edigarian and Wiegerinck, cf. [41]. Theproof will be an easy consequence of Theorem 4.4.1.

Theorem 5.3.1. [50] Let h : U −→ C be an F-holomorphic function on an F-open subset U of Cn. Then the zero set of h is pluripolar. In particular, the graphΓh(U) of h is also pluripolar.

Proof of Theorem 5.3.1. Let a ∈ U . Definition 5.2.2 gives us a compact (in theusual topology) F-neighborhood K of a in U , and a sequences (hn)n≥0, of holo-morphic functions defined in Euclidean neighborhoods of K such that

hn|K −→ h|K , uniformly.


For k ∈ N we define vn,k = max(log |hn|,−k) and vk = max(log |h|,−k). Clearly,vn,k converges uniformly on K to vk as n goes to infinity. Accordingly, vk isF-plurisubharmonic on the F-interior K ′ of K. Since vk is decreasing, the limitfunction log |h| is F-plurisubharmonic in K ′. Theorem 4.4.1 shows that the setK ′ ∩ h = 0 is pluripolar. The corollary follows now by application of the quasi-Lindelof property, cf. Theorem 3.2.5.

As an easy consequence, we recover the following result which was obtained in[41] first.

Corollary 5.3.2. Let f : U −→ Cn, f(z) = (f1(z), . . . , fn(z)), be a finely holo-morphic map on an F-open U ⊆ C. Then the graph Γf (U) of f over U is apluripolar subset of Cn+1.

Proof. It is enough to apply Theorem 5.3.1 to the function

h(z, w1, ..., wn) =

k=n∑

k=1

fk(z) − wk,

which is clearly F-holomorphic on the F-domain U × Cn.

Example 5.3.3. Let K ⊂ C be a compact set with non-empty F-interior K ′.Every function f ∈ R(K) (the uniform closure of the algebra of restrictions to Kof holomorphic functions in open sets containing K) is finely holomorphic in K ′,see Definition 2.4.1. Hence, by the above corollary, the graph Γf (K ′) = (z, f(z)) :z ∈ K ′ is a pluripolar subset of C2. Note however that in general K may nothave any Euclidean interior points. See also the examples in Chapter 7.

Remark 5.3.4. For n = 1, a partial converse of Corollary 5.3.2 was obtained byEdlund in his thesis [43]. Namely, he proved that if f is a function of class C2 onan F-open set V ⊂ C, and the graph Γf (V ) of f is pluripolar subset of C2, then fis finely holomorphic in V . Edlund’s result together with our Corollary 5.3.2 giveactually (in the particular case n = 1) a partial ”fine” analog of a deep theorem ofN. V. Shcherbina that was obtained shortly before, cf. [124]. Shcherbina’s resultsasserts that the graph Γf (Ω) of a continuous function f on an open set Ω ⊂ Cn ispluripolar subset of Cn+1 if and only if f is holomorphic. It is therefore a naturalquestion to ask whether the C2-regularity in Edlund’s theorem can be weakenedto just fine continuity. See also Section 7.3.

The inversion theorem for finely holomorphic functions, Theorem 2.4.14, allowsus to strengthen Corollary 5.3.2.

Corollary 5.3.5. Let f : U −→ Cn, f(z) = (f1(z), . . . , fn(z)), be a finely holo-morphic map on an F-domain U ⊆ C. Then the image f(U) of U is a pluripolarsubset of Cn.

The proof of Corollary 5.3.5 relies on Theorem 4.4.1 and the following lemmawhich appeared in [41].

5.3. Graphs of Finely Holomorphic Functions 73

Lemma 5.3.6. Let U ⊆ C be an F-domain, and let f : U −→ Cn, f(z) =(f1(z), . . . , fn(z)), be a finely holomorphic map. If h is a plurisubharmonic func-tion in Cn, Then the function h f is either finely subharmonic on U or ≡ −∞.

Proof. First, we assume that h is everywhere finite and continuous. Let a ∈ U .Definition 2.4.1 gives us a compact (in the usual topology) fine neighborhood Kof a in U , and n sequences (fk

j )k≥0, j = 1, . . . , n, of holomorphic functions definedin Euclidean neighborhoods of K such that

fkj |K −→ fj |K , j = 1, . . . , n uniformly.

Clearly, (fk1 , . . . , f

kn) converges uniformly on K to (f1, . . . , fn). Since h is con-

tinuous, the sequence h(fk1 , . . . , f

kn), of finite continuous subharmonic functions,

converges uniformly to h(f1, . . . , fn) on K. According to 4) in Theorem 2.3.7,h(f1, . . . , fn) is finely subharmonic in the fine interior of K.For the general case. We can assume that the fine interior of K is finely connected.Let (hm)m≥0 be a decreasing sequence of continuous plurisubharmonic functionswhich converges (pointwise) to h. By the first part of the proof, hm(f1, . . . , fn)is a decreasing sequence of finely subharmonic functions in the fine interior of K.The limit function h(f1, . . . , fn) is, by 3) in Theorem 2.3.7 , finely subharmonic oridentically −∞ in the fine interior of K. The sheaf property of finely subharmonicfunction, cf. [56, page 70] implies that h(f1, . . . , fn) is indeed finely subharmonicin all of U or is identically equal to −∞.

Proof of Corollary 5.3.5. Without loss of generality we may assume that f1 is notconstant and U is an F-domain. Let a ∈ U be such that the fine derivative f ′(a)is not equal to zero. By Theorem 2.4.14, there exists non-empty F-open subsetWa ⊆ U of U such that a ∈ Wa and f1|Wa : Wa −→ f1(Wa) is bijective and theinverse function f−1

1 is finely holomorphic in the F-open set f1(Wa). Now, observethat

f(Wa) = (w, f2(f−11 (w)), . . . , fn(f−1

1 (w))) : w ∈ f1(Wa),

where w = f1(z). Since the composition of two finely holomorphic functions isfinely holomorphic, cf. Corollary 2.4.10, the map

w 7→ (f2(f−11 (w)), . . . , fn(f−1

1 (w)))

is finely holomorphic in f1(Wa). By Corollary 5.3.2, the graph

(w, f2(f−11 (w)), . . . , fn(f−1

1 (w))) : w ∈ f1(Wa) = f(Wa)

is a pluripolar subset of Cn. Josefson’s theorem ensures the existence of a pluri-subharmonic function h ∈ PSH(Cn) such that

h(f1(z), . . . , fn(z)) = −∞, ∀z ∈Wa.

By Lemma 5.3.6, the function z 7→ h(f1(z), . . . , fn(z)) is either finely subharmonicor identically equal to −∞. Since it equals −∞ on the non-polar subset Wa ⊂U , we must have h(f(U)) = −∞, by Theorem 2.3.14 (or Theorem 4.4.1). Thiscompletes the proof.


5.4 Application to Pluripolar Hulls

Theorem 5.4.1. Let f : U −→ Cn, f(z) = (f1(z), ..., fn(z)), be a finely holo-morphic map on an F-domain U ⊆ C. If E is a non-polar subset of U , then thepluripolar hull of f(E) contains f(U).

Proof. By Corollary 5.3.5 , the set f(E) is a pluripolar subset of Cn. Let ϕ ∈PSH(Cn) be such that ϕ 6≡ −∞, and ϕ(f(E)) = −∞. By Lemma 5.3.6, thefunction ϕf is finely subharmonic on U or identical to −∞. Since it assumes thevalue −∞ on the non-polar set E, we must have ϕ f ≡ −∞ on U , by Theorem2.3.14 (or Theorem 4.4.1). Hence f(U) ⊂ (f(E))∗

Cn

As a particular case of Theorem 5.4.1, we formulate here a generalized andprecise version of Theorem 5.1.1.

Corollary 5.4.2. Let f be holomorphic in a connected open set U ⊂ C and letp ∈ ∂U . Suppose that f has a finely holomorphic continuation F at p to an F-openand F-connected neighborhood V of p. Then ΓF (V ) ⊂ (Γf (U))∗

C2 . Moreover, if E

is a non-polar subset of V ∩ U then Γf (U) ∪ ΓF (V ) ⊂ (Γf (E))∗C2 .

Proof. Denote by g the finely holomorphic function which is equal to f on Uand to F on V . Let h ∈ PSH(C2) be a plurisubharmonic function such thath(z, f(z)) = −∞, ∀z ∈ U . According to Lemma 5.3.6, the function z → h(z, g(z))is finely subharmonic on the F-domain U ∪ V or ≡ −∞. Since it assumes −∞on the non-polar set U , it must be identically equal to −∞ in view of Theorem2.3.14 (or Theorem 4.4.1). Hence ΓF (V ) ⊂ (Γf (U))∗

C2 . The second statement canbe proved similarly. See Proposition 5.5.1 below for a more general results.

The next theorem is a higher dimensional analog of Theorem 5.4.1, cf. [50].

Theorem 5.4.3. Let U ⊂ Cn be an F-domain, and let h be F-holomorphic in U .Denote by Γh(U) the graph of h over U , and let E be a non-pluripolar subset ofU . Then Γh(U) ⊂ (Γh(E))∗

Cn+1 .

To prove Theorem 5.4.3 we need the following lemma, which is the higherdimensional version of Lemma 5.3.6 with a similar proof.

Lemma 5.4.4. Let U ⊆ Cn be an F-domain, and let f : U −→ C be an F-holomorphic function. Suppose that h : Cn+1 −→ [−∞, +∞[ is a plurisub-harmonic function. Then the function z 7→ h(z, f(z)) is F-plurisubharmonic onU .

Proof. First, we assume that h is continuous and finite everywhere. Let a ∈ U . ByDefinition 5.2.2 there is a compact F-neighborhood K of a in U and a sequencefk of holomorphic functions defined in usual neighborhoods of K that convergesuniformly to f |K . Since h(z, fk(z)) is plurisubharmonic and converges uniformlyto h(z, f(z)) on K, h(z, f(z)) is F-plurisubharmonic in the F-interior of K. Now

5.4. Application to Pluripolar Hulls 75

by the sheaf property of F-plurisubharmonic function we conclude that h(z, f(z))is F-plurisubharmonic on U .Suppose that h is arbitrary. Then h is the limit of some decreasing sequence ofcontinuous plurisubharmonic functions hn ∈ PSH(Cn+1). By the first part of theproof, hn(z, f(z))n is a decreasing sequence of F-plurisubharmonic functions inthe F-interior of K. The limit h(z, f(z)) is therefore F-plurisubharmonic in theF-interior of K.

Proof of Theorem 5.4.3. By Theorem 5.3.1, The set Γh(E) is pluripolar subset ofCn+1. Let ϕ ∈ PSH(Cn+1) with ϕ 6≡ −∞ and ϕ(z, h(z)) = −∞, for every z ∈ E.Lemma 5.4.4 shows that the function z 7→ ϕ(z, h(z)) is F-plurisubharmonic in U .Since E is not pluripolar, it follows from Theorem 4.4.1 that ϕ(z, h(z)) = −∞everywhere in U . Hence Γh(U) ⊂ (Γh(E))∗

Cn+1 .

Theorem 5.4.3 gives, in particular, an extension of Theorem 5.1.1 to functionsof several complex variables. For convenience of the reader we formulate here,without proof, the precise version.

Corollary 5.4.5. Let f be holomorphic in a connected open set U ⊂ Cn and letp ∈ ∂U . Suppose that f has an F-holomorphic continuation F at p to an F-openand F-connected neighborhood V of p. Then ΓF (V ) ⊂ (Γf (U))∗

C2 . Moreover, if E

is a non-polar subset of V ∩ U then Γf (U) ∪ ΓF (V ) ⊂ (Γf (E))∗C2 .

Remark 5.4.6. Corollary 5.4.5 only explains for a small part the propagationof pluripolar hulls. E.g., take U the unit ball in Cn and consider the functiong(z) = f(z)(z1 − z2

2). Then, whatever the extendibility properties of f may be,the pluripolar hull of the graph of g will contain the set z1 = z2

2.

The next theorem is a simple, precise, and complete interpretation of recentresults of Edigarian and Wiegerinck, cf. [38, 40].

Theorem 5.4.7. Let D be a domain in C and let A be a closed polar subset ofD. Suppose that f ∈ O(D\A) and that z0 ∈ A. Then the following conditions areequivalent:(1) (z0 × C) ∩ (Γf )∗D×C

6= ∅.

(2) f has a finely holomorphic extension f at z0.Moreover, if one of these conditions is met, then (z0×C)∩(Γf )∗D×C

= (z0, f(z0)).

Proof. (1) ⇒ (2). According to Theorem 5.2.1, there exists R > 0 such that theset z ∈ D\A : |f(z)| ≥ R is thin at z0. Clearly, the set U = z ∈ D\A :|f(z)| < R ∪ z0 is an F-open neighborhood of z0. Since f is bounded inU\z0 and finely holomorphic in U\z0, Corollary 2.4.5 shows that f has afinely holomorphic extension at z0.(2) ⇒ (1). Suppose that f has a finely holomorphic extension f at z0. Clearly,(D\A) ∪ z0 is an F-open neighborhood of z0. Since polar sets do not separateF-domains, cf. Theorem 2.2.29, the set (D\A) ∪ z0 is F-connected. Let h ∈PSH(D × C) be a plurisubharmonic function such that h(z, f(z)) = −∞, ∀z ∈D\A. According to Lemma 5.3.6, the function z 7→ h(z, f(z)) is either finely


subharmonic on (D\A) ∪ z0 or ≡ −∞. As it assumes −∞ on D\A, it must beidentically equal to −∞ in view of Theorem 2.3.14. Consequently, (z0, f(z0)) ∈(Γf )∗D×C

. The last assertion follows from Theorem 5.10 in [40].

5.5 Concluding Remarks and Open Questions

A finely analytic curve is a pair (U, f), where U is an F-domain in C and f =(f1, . . . , fn) : U → Cn is a finely holomorphic map. As usual we will identify acurve with its image.

Let E ⊂ Cn be a pluripolar set and E∗Cn its pluripolar hull. It follows from

the arguments used before that if E hits a finely analytic curve f(U) in somenon ”small” set, then E∗

Cn contains all the points of f(U). Namely, we have thefollowing.

Proposition 5.5.1. Let f : U −→ Cn be a finely holomorphic map on a boundedF-domain U ⊂ C and let E ⊂ Cn be a pluripolar set. If f(U) ∩ E 6= ∅ andf−1(f(U) ∩ E) is non-polar, then f(U) ⊂ E∗

Cn .

Proof. Let h ∈ PSH(Cn) be a plurisubharmonic function such that h(z) = −∞,∀z ∈ E. By Lemma 5.3.6, h f is either finely subharmonic on U or ≡ −∞. Asit assumes −∞ on f−1(f(U)∩E), it must be, by Theorem 2.3.14, identically −∞on U . This proves that f(U) ⊂ E∗

Cn .

The conclusion of the above proposition remains valid if one assumes that Econtains merely the “boundary of a finely analytic curve”.

Proposition 5.5.2. Let f and E be as above. If f extends by fine continuity tothe F-boundary ∂fU of U and f(∂fU) ⊂ E, then f(U) ⊂ E∗

Cn .

Proof. Let h ∈ PSH(Cn) be plurisubharmonic function such that h(z) = −∞,∀z ∈ E. Let a ∈ ∂fU . By assumption, f has a fine limit at a. Using Cartan’stheorem (cf. [76], Theorem 10.15), one can easily find a finely open neighborhoodVa of a such that the usual limit, limz→a,z∈Va∩U f(z), exists and is equal to f(a).Let M > 0. Since h is upper semicontinuous, the set z ∈ Cn : h(z) < −M isopen. As f(a) ∈ z ∈ Cn : h(z) < −M, one can find a positive number δa > 0such that

f(w) ∈ z ∈ Cn : h(z) < −M, ∀w ∈ D(a, δa) ∩ Va ∩ U,

where D(a, δa) is the disk with center a and radius δa. Consequently

f- lim supz→a,z∈U

h(f(z)) ≤ f- lim supz→a,z∈Va∩U

h(f(z)) < −M, ∀a ∈ ∂fU

where f- lim sup denotes the limit with respect to the fine topology.By Lemma 5.3.6, the function h f is a finely subharmonic function on U , or

is identically equal to −∞. the fine boundary maximum principle, cf. Theorem2.3.11, shows that h f(z) < −M , ∀z ∈ U . Since M was arbitrary, we concludethat h f(U) = −∞. This proves the proposition.

5.5. Concluding Remarks and Open Questions 77

Our results reveal a very close relationship between the pluripolar hull of thegraph of a holomorphic function and the theory of finely holomorphic functions(see also [44]). This leads naturally to the following fundamental problem.Problem 1. Let f : Ω −→ C be a holomorphic function on a simply connectedopen subset Ω ⊂ C. Suppose that the graph Γf (Ω) of f over Ω is not completepluripolar. Must then (Γf (Ω))∗

C2\Γf (Ω) have a fine analytic structure? i.e., letz ∈ (Γf (Ω))∗

C2\Γf (Ω). Must there exist a finely analytic curve passing through zand contained in (Γf (Ω))∗

C2\Γf (Ω)?Obviously, a positive answer to the above problem would, in particular, solve thefollowing problem posed in [44].Problem 2. Let f be a holomorphic function in the unit disk D. Suppose that(Γf (D))∗

C2 is the graph of some function F . Is F a finely holomorphic continuationof f?

It was proved in [25] that one can not detect ”pluripolarity” via intersectionwith one dimensional complex analytic varieties. Since there are, roughly speaking,much more finely analytic curves in Cn than analytic varieties, one can naturallypose the followingProblem 3. Let K be a compact set in Cn and suppose that f−1(K ∩ f(U)) is apolar subset of U (or empty) for any finely analytic curve f : U −→ Cn. Must Kbe a pluripolar subset of Cn?

Chapter 6

Fine Analytic Structure

This chapter is joint work with Tomas Edlund and presents, with no changes, thecontents of the paper [46] which will appear in Indagationes Mathematicae.

We discuss the relation between pluripolar hulls and fine analytic structure.Our main result is the following. For each non polar subset S of the complexplane C we prove that there exists a pluripolar set E ⊂ (S×C) with the propertythat the pluripolar hull of E relative to C2 contains no fine analytic structure andits projection onto the first coordinate plane equals C.

6.1 Introduction

Denote by Ω an open subset of Cn and let E ⊂ Ω be a pluripolar subset. Itmight be the case that any plurisubharmonic function u(z) defined in Ω that isequal to −∞ on the set E is necessarily equal to −∞ on a strictly larger set. Forinstance, if E contains a non polar proper subset of a connected Riemann surfaceembedded into Cn, then any plurisubharmonic function defined in a neighborhoodof the Riemann surface which is equal to −∞ on E is automatically equal to −∞on the whole Riemann surface. In order to try to understand some aspect of theunderlying mechanism of the described ”propagation” property of pluripolar sets,the pluripolar hull of graphs Γf (D) of analytic functions f in a domain D ⊂ C hasbeen studied in a number of papers. (See for instance [38], [44], [93] and [134].)

The pluripolar hull E∗Ω relative to Ω of a pluripolar set E is defined as follows.

E∗Ω =

⋂

z ∈ Ω : u(z) = −∞,

where the intersection is taken over all plurisubharmonic functions defined in Ωwhich are equal to −∞ on E. The set E is called complete pluripolar in Ω if thereexists a plurisubharmonic function on Ω which equals −∞ precisely on E.

As remarked above a necessary condition for a pluripolar set E to satisfyE∗

Ω = E is that E ∩ A is polar in A (or E ∩ A = A) for all one-dimensionalcomplex analytic varieties A ⊂ Ω. The fact that this is not a sufficient condition

79

80 Chapter 6. Fine Analytic Structure

was proved by Levenberg in [91]. By using a refinement of Wermer’s example of apolynomially convex compact set with no analytic structure (cf. [132]) Levenbergproved that there exists a compact set K ⊂ C2 satisfying K 6= K∗

C2 , and theintersection of K with any one dimensional analytic variety A is polar in A. Inthis example it is not clear what the pluripolar hull K∗

C2 equals.We will say that a set S ⊂ Cn contains fine analytic structure if there exists

a non constant map ϕ : U → S from a fine domain U ⊂ C whose coordinatefunctions are finely holomorphic in U (see Definition 2.3 below). Such a map ϕwill be called a finely analytic curve.

Motivated by recent results of Joricke and the first author (cf. [44]), the fol-lowing result was proved in [41].

Theorem 6.1.1. Let ϕ : U −→ Cn be a finely holomorphic map on a fine domainU ⊂ C and let E ⊂ Cn be a pluripolar set. Then the following hold(1) ϕ(U) is a pluripolar subset of Cn

(2) If ϕ−1(ϕ(U) ∩ E) is a non polar subset of C then ϕ(U) ⊂ E∗Cn .

In view of this result one may expect to get more information on the pluripolarhull E∗

Cn by examining the intersection of the pluripolar set E with finely analyticcurves. Since many curves in Cn are complete pluripolar (see [45]) one cannotexpect that E∗

Cn always contains fine analytic structure. However if we considerthe non trivial part E∗

Cn rE the situation is up to now slightly different. In fact,all examples we have seen so far have the property that if E∗

Cn r E is nonemptythen for each w ∈ E∗

Cn rE there exists a finely analytic curve ϕ contained in E∗Cn

which passes through the point w. (i.e. ϕ : U → E∗C2 is a finely analytic curve and

ϕ(z) = w for some z ∈ U). In this paper we prove that no such conclusion holdsin general. We have the following main result.

Theorem 6.1.2. For each proper non polar subset S ⊂ C there exists a pluripolarset E ⊂ (S×C) with the property that E∗

C2 contains no fine analytic structure andthe projection of E∗

C2 onto the first coordinate plane equals C.

The set E will be a subset of a complete pluripolar set X which is constructedin the same spirit as Wermer’s polynomially convex compact set without analyticstructure.

Let us describe more precisely the content of the paper. In Section 2 we brieflyrecall the construction of Wermer’s set and prove that it contains no fine analyticstructure. This leads to Theorem 6.2.4 which slightly generalizes a result in [91].The main result is proved in Section 3. Subsection 3.1 is devoted to constructthe above mentioned set X and in Subsection 3.2 we show that X contains nofine analytic structure. In Subsection 3.3 we define the set E and describe E∗

C2 .Finally, in Section 4 we make some remarks and pose two open questions.

Readers who are not familiar with basic results on finely holomorphic functionsand fine potential theory are referred to [56] and [61].Acknowledgments. Part of this work was completed while the first author wasvisiting Korteweg-de Vries Institute for Mathematics, University of Amsterdam.

6.2. Wermer’s Example 81

He would like to thank this institution for its hospitality. The authors thankProfessor Jan Wiegerinck for very helpful discussions.

6.2 Wermer’s Example

In this Section we sketch the details of Wermer’s construction given in [132]. De-note by Dr the open disk with center zero and radius r and by Cr the open cylinderDr×C. Let a1, a2, .... denote the points in the disk D 1

2with rational real and imag-

inary part. For each j we denote by Bj(z) the algebraic (2-valued) function

Bj(z) = (z − a1)(z − a2)...(z − aj−1)√

(z − aj).

To each n-tuple of positive constants c1, c2, ..., cn we associate the algebraic (2n-valued) function gn(z) =

∑nj=1 cjBj(z). Let

∑

(c1, ..., cn), n = 1, 2, ... be the

subset of the Riemann surface of gn(z) which lies in C 12.

Lemma 6.2.1. [[132], lemma 1] There exist positive constants εn and cn, n =1, 2..., with c1 = 1

10 and cn+1 ≤ ( 110 )cn, n = 1, 2, ... and a sequence of polynomials

pn(z, w) such that:(1) pn = 0 ∩ |z| ≤ 1

2 =∑

(c1, ..., cn), n = 1, 2, ...(2) |pn+1| ≤ εn+1 ∩ |z| ≤ 1

2 ⊂ |pn| ≤ εn ∩ |z| ≤ 12, n = 1, 2, ...

(3) If |a| ≤ 12 and |pn(a, w)| ≤ εn, then there is a wn with pn(a, wn) = 0 and |w−

wn| ≤1n , n = 1, 2, ....

With pn, εn, n = 1, 2, ... chosen as in Lemma 6.2.1, we put

Y =∞⋂

n=1

[|pn| ≤ εn ∩ |z| ≤1

2].

Clearly, Y is a compact polynomially convex subset of C2. It was shown by Wermerthat Y has no analytic structure i.e. Y contains no non-constant analytic disk. Infact he proves something stronger. The set Y defined above contains no graph ofa continuous function defined on a circle in D 1

2which avoids all the branch points

ai. Using this observation the following lemma follows.

Lemma 6.2.2. There is no finely analytic curve contained in Y .

Before we prove Lemma 6.2.2 we recall the following definition (cf. [61], page75):

Definition 6.2.3. Let U be a finely open set in C. A function f : U −→ C is saidto be finely holomorphic if every point of U has a compact (in the usual topology)fine neighborhood K ⊂ U such that the restriction f |K belongs to R(K).

Here R(K) denotes the uniform closure of the algebra of all restrictions to Kof rational functions on C with poles off K.


Proof of Lemma 2.2. Let ϕ : U → Y , z 7→ (ϕ1(z), ϕ2(z)) be a finely analytic curvecontained in Y . If ϕ1(z) is constant on U then ϕ2(z) must also be a constantsince non constant finely holomorphic functions are finely open maps and by theconstruction of the set Y the fibre Y ∩ (z×C) is a Cantor set or a finite set forany point z ∈ D1/2. Assume therefore that ϕ1(z) is non-constant. In particular,there is a point z0 ∈ U where the fine derivative of ϕ1(z) does not vanish. Henceϕ1(z) is one-to-one on some finely open neighborhood V ⊂ U of the point z0. Byconsidering the map z 7→ (ϕ1 ϕ

−11 (z), ϕ2 ϕ

−11 (z)), defined on the finely open set

ϕ1(V ) we may assume that ϕ is of the form z 7→ (z, g(z)) where g(z) = ϕ2 ϕ−11 (z)

is finely holomorphic in the finely open set V ′ = ϕ1(V ) ⊂ D1/2. By Definition6.2.3 there exists a compact subset K ⊂ V ′ with non-empty fine interior suchthat g(z) is a continuous function on K (with respect to the Euclidean topology).ShrinkingK if necessary we may assume thatK∩a1, a2, .... = ∅. Let p be a pointin the fine interior of K. It is well known that there exists a sequence of circlesC(p, rj) contained in K with centers p and radii rj → 0 as j → ∞. Clearly,the circle C(p, rj) avoids the branch points a1, a2, .... and its image under thecontinuous map z 7→ (z, g(z)) is contained in Y . By the above observation this isnot possible. Hence Y contains no fine analytic structure.

Denote by dn the degree of the one variable polynomial w 7→ pn(z, w) wherepn(z, w) is the polynomial given in Lemma 6.2.1. Assume that the set Y is con-structed using the parameters εn satisfying the following condition

limn→∞

(εn)1/dn = 0. (6.2.1)

It is shown in [92] that with this choice the set Y ∩ C1/2 is complete pluripolar inC1/2. Using this result and Lemma 6.2.2 we are able to generalize a result in [91].

Theorem 6.2.4. Fix δ ∈ (0, 1/2) and let Yδ =⋂∞

n=1[|pn| ≤ εn ∩ |z| ≤ δ] beconstructed using the parameters εn satisfying (6.2.1). Then(a) ϕ−1(ϕ(U)∩Yδ) is a polar subset of U for all finely analytic curves ϕ : U → C2.(b) Yδ 6= (Yδ)

∗C2 .

Proof of Theorem 6.2.4. In order to prove (a) we argue by contradiction. Assumetherefore that ϕ : U → C2 is a finely analytic curve and ϕ−1(ϕ(U) ∩ Yδ) is a nonpolar subset of U . Then there is a fine domain Uk0 ⊆ U such that ϕ(Uk0) ⊂ C1/2

and ϕ−1(ϕ(Uk0)∩Yδ) is non polar. Indeed, the set ϕ−1(ϕ(U)∩C1/2) is a finely opensubset of U and hence has at most countably many finely connected componentsUk∞k=1. Moreover, ϕ−1(ϕ(U) ∩ Yδ) ∩ Uk0 is non polar for some natural numberk0, since otherwise

⋃∞k=1ϕ

−1(ϕ(U)∩ Yδ)∩Uk = ϕ−1(ϕ(U)∩Yδ) would be polarcontrary to our assumption. Since Y ∩ C1/2 is complete pluripolar in C1/2 thereexists a plurisubharmonic function u defined in C1/2 which is equal to −∞ exactlyon Y ∩ C1/2. By Lemma 5.3.6, the function u ϕ is either finely subharmonicon Uk0 or identically equal to −∞.. Since u equals −∞ on the non polar subsetϕ−1(ϕ(U) ∩ Yδ) ∩ Uk0 , it must be identically equal to −∞ on Uk0 . Thereforeϕ(Uk0) ⊂ u = −∞ = Y ∩ C1/2 contradicting Lemma 6.2.2 and (a) follows.

6.3. Proof of Theorem 6.1.2 83

The proof of assertion (b) follows immediately from the proof of Proposition 3.1in [91]. Indeed, if u is a plurisubharmonic function defined in C2 which equals −∞on Yδ then the function z 7→ maxu(z, w) : (z, w) ∈ Y is subharmonic in D1/2 andsince it equals −∞ on Dδ it equals −∞ on D1/2. Consequently Y ∩ C1/2 ⊂ (Yδ)

∗C2

and hence Yδ 6= (Yδ)∗C2 .

Remark. It follows from the argument used in the proof of assertion (b) inTheorem 6.2.4 that Y ∩ C1/2 ⊂ (Yδ)

∗C1/2

. Since the first set is complete pluripolar

in C1/2 it follows that (Yδ)∗C1/2

= Y ∩ C1/2. Consequently, (Yδ)∗C1/2

contains no

fine analytic structure. It would be nice to determine what the set (Yδ)∗C2 equals

and to figure out whether this set contains fine analytic structure. We are unableto do this. But by modifying Wermer’s construction, we will in the next Sectionconstruct a complete pluripolar Wermer-like set X ⊂ C2 with the property that(X ∩ (S×C))∗

C2 contains no fine analytic structure for all non polar subset S ⊂ C.

6.3 Proof of Theorem 6.1.2

6.3.1 Construction of the Set X

In this Subsection we construct the set X . Denote by ak∞k=1 the points in thecomplex plane both of whose coordinates are rational numbers. Without loss ofgenerality we may assume that ak ∈ Dk. For any sequence of points al

jl=1 we

denote by Bj(z) the algebraic function

Bj(z) = (z − a1) . . . (z − aj−1)√

(z − aj).

Denote by γj a simple smooth curve with endpoints aj and ∞. For each j, Bj(z)has two single-valued analytic branches on C r γj . Following the notation in [132]we choose one of the branches Bj(z) arbitrarily and denote it by βj(z). Then|βj(z)| = |Bj(z)| is continuous on C.

For each n+1-tuple of positive constants (c1, c2, . . . , cn+1) we denote by gn(z)the algebraic function defined recursively in the following way. Put g1(z) =c1B1(z) and g2(z) = c1B1(z) + c2B2(z) and if gn(z) has been chosen we willchoose gn+1(z) as described below. Put Z1(z) = 1 and for n = 2, 3, . . . define thefunction Zn(z) as follows. Denote by z1, z2, . . . , zl all the zeros of all possible differ-ent differences hj(z)− hi(z) (i 6= j) of branches hi(z), hj(z) of the function gn(z).Suppose zk is a zero of hj(z)−hi(z) of order mk and put Zn(z) = Πl

i=1(z− zi)mi .

Note that the zeros of Zn(z) are also zeros of the function Zn+1(z) of the same orgreater multiplicity. Define gn+1(z) = gn(z) + cn+1Zn(z)Bn+1(z).

By Σ(c1, . . . , cn) we mean the Riemann surface of gn(z) which lies in C2. Inother words, Σ(c1, . . . , cn) = (z, w) : z ∈ C, w = wj , j = 1, 2, . . . , 2n, wherewj , j = 1, 2, . . . , 2n are the values of gn(z) at z.


We will choose positive constants cn, εn and polynomials pn(z, w) recursivelyso that

pn(z, w) = 0 ∩ Cn+1 = Σ(c1, c2, . . . , cn) ∩ Cn+1 and (6.3.1)

|pn+1(z, w)| ≤ εn+1 ∩ Cn+1 ⊂ |pn(z, w)| < εn ∩ Cn+1 (6.3.2)

hold for n = 1, 2, . . . . The set X will be of the form

X =

∞⋃

n=1

(

∞⋂

j=n

|pj(z, w)| ≤ εj ∩ Cn+1

)

. (6.3.3)

Put c1 = 1 and let p1(z, w) = w2 − (z − a1). It is clear that Σ(c1) ∩ C2 =p1(z, w) = 0 ∩ C2. Choose ε1 > 0 so that if z0 ∈ D2 and |p1(z0, w)| ≤ ε1 thenthere exists (z0, w1) ∈ Σ(c1)∩C2 with |w−w1| ≤ 1. Let B2 = D2×Dρ1 be a bidiskwhere ρ1 is chosen so that

|p1(z, w)| ≤ ε1 ∩ C2 = |p1(z, w)| ≤ ε1 ∩ B2.

Assume that cn, εn and pn(z, w) have been chosen so that (6.3.1) and (6.3.2)hold. We will now choose cn+1 and pn+1(z, w). We denote by wj(z), j =1, 2, . . . , 2n the roots of pn(z, ·) = 0 and to each positive constant c we assigna polynomial pc(z, w) by putting

pc(z, w) = Π2n

j=1

(

(w − wj(z))2 − c2(Zn(z)Bn+1(z))

2)

. (6.3.4)

Then pc(z, ·) = 0 has the roots wj(z) ± cZn(z)Bn+1(z), j = 1, 2, . . . , 2n and so

pc(z, w) = 0 = Σ(c1, c2, . . . , cn, c).

Note that from (6.3.4)

pc = p2n + c2q1 + ...+ (c2)2

n

q2n ,

where the qj are polynomials in z and w, not depending on c. Choose c > 0 sothat the following hold for all z ∈ Dn+1.

Σ(c1, c2, . . . , cn, c) ∩ Cn+1 ⊂ |pn(z, w)| < εn/2 ∩ Cn+1and (6.3.5)

c · |Zn(z)Bn+1(z)| ≤ (1/10)cn|Zn−1(z)Bn(z)|. (6.3.6)

Decreasing c if necessary we may assume that if hi(z) and hj(z) are any differentbranches of the function gn(z) the estimate

|hj(z) − hi(z)| ≥ 2c|Zn(z)Bn+1(z)| (6.3.7)

holds in Dn+1 with equality exactly at the zeros of Zn(z) which are contained inDn+1 and at the points a1, . . . an. This estimate will be needed later when weprove that X contains no fine analytic structure. Choose cn+1 = c.


Let Bn+2 = Dn+2×Dρn+2 be a bidisk where ρn+2 is chosen so that |pn(z, w)| ≤εn∩Cn+2 = |pn(z, w)| ≤ εn∩Bn+2 and ρn+2 > ρn+1+1. Let δ > 0 be a constantsuch that |δ · pc(z, w)| < 1 in Bn+2 and choose pn+1(z, w) = δ · pc(z, w).

We now turn to the choice of εn+1. Since the part of the zero set of pn+1(z, w)which is contained in Bn+1 is a subset of |pn(z, w)| < εn/2 ∩ Bn+1 it is possibleto find a natural number mn+1 so that

1

mn+1log |pn+1(z, w)| ≥ −

1

2nfor all (z, w) ∈ Bn+1 r |pn(z, w)| ≤ εn. (6.3.8)

Choose εn+1 < εn so that

1

mn+1log |pn+1(z, w)| ≤ −1 for all (z, w) ∈ |pn+1(z, w)| ≤ εn+1 ∩ Cn+2. (6.3.9)

By decreasing εn+1 we may assume that (6.3.2) and the following assumption hold.

If (z0, w) ∈ Cn+2 and |pn+1(z0, w)| ≤ εn+1, then there exists

(z0, wn) ∈ Cn+2 such that |pn+1(z0, wn)| = 0 and |w − wn| ≤ 1/n.(6.3.10)

This ends the recursion.

Lemma 6.3.1. The set X defined by (6.3.3) is complete pluripolar in C2.

Proof. Define for n ≥ 2 the plurisubharmonic function

un(z, w) = max 1

mnlog |pn(z, w)|,−1

and put u(z, w) =∑

n≥2 un(z, w). Then u(z, w) is plurisubharmonic in C2. In-

deed, since the bidisks Bn exhaust C2 and |pn(z, w)| < 1 in Bn+1 the series∑

n≥2 un(z, w) will be decreasing on each fixed bidisk BN after a finite num-ber of terms and hence plurisubharmonic there. Since plurisubharmonicity isa local property u(z, w) is plurisubharmonic in C2. If (z0, w0) ∈ X , then forsome natural number N , (z0, w0) ∈

⋂∞j=N|pj(z, w)| ≤ εj ∩ CN+1. Condition

(6.3.9) above implies that u(z0, w0) = Const +∑

n>N un(z0, w0) = −∞. Finallyif (z0, w0) /∈ X then there exists a natural number N such that (z0, w0) ∈ BN and(z0, w0) /∈ |pn(z, w)| ≤ εn ∩ BN for all n ≥ N . By (6.3.8)

u(z, w) = Const+∑

n>N

max 1

mnlog |pn(z, w)|,−1

≥ Const+∑

n>N

−1

2n> −∞.

The Lemma follows.

6.3.2 X Contains No Fine Analytic Structure

In this Section we show that X contains no fine analytic structure. Suppose thatz 7→ (ϕ1(z), ϕ2(z)) is a finely analytic curve whose image is contained in X . If


ϕ1(z) is constant then ϕ2(z) must be constant since X ∩ (z0×C) is a Cantor setor a finite set for any point z0 ∈ C. On the other hand, if ϕ1(z) is non-constant,then using the arguments given in the proof of Lemma 6.2.2 we may assume thatthe finely analytic curve contained in X is given by z 7→ (z,m(z)) where m(z) is afinely holomorphic function defined in U where U ⊂ Dn for some natural numbern. Fix a point z′ ∈ U r a1, . . . , an. By the definition of finely holomorphicfunctions we can find a compact (in the usual topology) fine neighborhood K ⊂ Uof z′ where m(z) is continuous. Shrinking K if necessary we may assume that(K r z′) ∩ (aj∞j=1 ∪ Zk−1(z) = 0∞k=2) = ∅. Since the complement of Kis thin at z′, one can find a sequence of circles C(z′, ri) ⊂ K with ri → 0 asi→ ∞. Choose one of the circles C(z′, rj) so that none of the points a1, . . . , an arecontained in |z − z′| ≤ rj. Let ak be the first point in the sequence aj∞j=n+1

which is contained in |z − z′| ≤ rj. Note that ak ∈ |z − z′| < rj, m(z) iscontinuous on C(z′, rj) and the function Zk−1(z)βk(z) 6= 0 when z ∈ C(z′, rj).The fact that the image of C(z′, rj) under the map z 7→ (z,m(z)) is a subset of Xwill lead us to a contradiction and hence X contains no fine analytic structure. Inorder to prove this fix a point z1 ∈ C(z′, rj) and denote by < the 2k branches ofthe algebraic function gk(z) defined on C(z′, rj) r z1.

Lemma 6.3.2. If hi(z) and hj(z) are any different functions from <, then

|hi(z) − hj(z)| > (3/2)ck|Zk−1(z)βk(z)| (6.3.11)

holds for all z ∈ C(z′, rj) r z1.

Proof. This follows directly from (6.3.7) since C(z′, rj) ⊂ Dn and C(z′, rj) doesnot intersect any of the branch points a1, . . . , ak or the zeros of Zk−1(z).

From now on the proof that X contains no fine analytic structure follows thearguments given in [132].

Lemma 6.3.3. Fix z0 in C(z′, rj) r z1. There exists a function hi(z) ∈ <,where hi(z) depends on z0 such that

|m(z0) − hi(z0)| < (1/4)ck|Zk−1(z0)βk(z0)| (6.3.12)

Proof. By (6.3.10) there exists N ≥ k and wN such that (z0, wN ) lies on the setΣ(c1, . . . , cN ) and m(z0) = wN +R(z0) where |R(z0)| ≤ (1/10)ck|Zk−1(z0)βk(z0)|.Thus

m(z0) = ±c1β1(z0) +N∑

ν=2

±cνZν−1(z0)βν(z0) +R(z0) =

def= hi(z0) +

N∑

ν=k+1

cνZν−1(z0)βν(z0) +R(z0).


6.3.3 The Sets E and E∗

C2

Denote by E the pluripolar set E = (S ×C)∩X where S is a non polar subset ofC. Since X is complete pluripolar in C2 it follows that E∗

C2 ⊂ X . To prove thatX ⊂ E∗

C2 we argue as follows. First we claim that the set X is pseudoconcave.Indeed, by the construction of the set X ,

C2 rX = ∪∞n=1|pn(z, w)| > εn ∩ Cn+1. (6.3.16)

By the choice of the polynomials pn(z, w) it follows that

|pn(z, w)| > εn ∩ Cn+1 ⊂ |pn+1(z, w)| > εn+1 ∩ Cn+2.

Moreover, for each natural number n the set |pn(z, w)| > εn ∩ Cn+1 is a domainof holomorphy. Hence C2 r X is a countable union of increasing domains ofholomorphy. By the Behnke-Stein Theorem C2 r X is pseudoconvex and theclaim follows.

Denote by u(z, w) a globally defined plurisubharmonic function which equals−∞ on E. It is shown in [127] that the function z 7→ maxu(z, w) : (z, w) ∈ Xis subharmonic in C. Since the projection S of E onto the first coordinate planeis non polar the function z 7→ maxu(z, w) : (z, w) ∈ X will be identically equalto −∞ on C hence u(z, w) = −∞ on the whole of X and consequently E∗

C2 = X .This ends the proof of Theorem 6.1.2.

6.4 Final Remarks and Open Problems

It follows immediately from Theorem 6.1.1 and the fact that X contains no fineanalytic structure that if ϕ : U → C2 is a finely analytic curve, then the setϕ−1(ϕ(U) ∩X) is polar in C.

Despite the result of Theorem 6.1.2 it should be mentioned here that in the sit-uation where one considers the pluripolar hull of the graph of a finely holomorphicfunction defined in a fine domain D, the following problem still remains open.Problem 1. Let z ∈ Γf (D)∗

C2 . Does this imply that there is a finely analyticcurve contained in Γf (D)∗

C2 which passes through the point z?It is proved in [40] that the pluripolar hull relative to Cn of a connected pluri-

polar Fσ subset is a connected set. It is a fairly easy exercise to show that the setX = E∗

C2 in Theorem 6.1.2 is path connected, but in general the pluripolar hull ofa connected (Fσ) pluripolar set is not path connected. Indeed, denote by f(z) anentire function of order 1/3. f(1/z) has an essential singularity at 0 and in [134]Wiegerinck proved that the graph Γf(1/z) of f(1/z) over C r 0 is complete plu-ripolar in C2. Consequently, if we put E = Γf(1/z) ∪ (0×C) then E is completepluripolar in C2 and hence E∗

C2 = E. Moreover E is a connected Fσ subset of C2.By the famous Denjoy-Carleman-Ahlfors theorem (see e.g. [1]), entire functions oforder 1/3 do not have finite asymptotic values; i.e., there are no curves γ endingat infinity such that f(z) approaches a finite value as z → ∞ along γ. Hence it isnot possible to find a path in E∗

C2 connecting a point on Γf(1/z) with a point in

6.4. Final Remarks and Open Problems 89

the set 0×C. In view of this remark it would be interesting to know the answerto the following question.Problem 2. Is Γf (D)∗

C2 path connected ?

Chapter 7

Examples and Open

Questions

In this last chapter we study the pluripolar hull of graphs of a Borel series. Thefirst example shows that some of the techniques used in [38] can be carried overto the case of a graph of a continuous functions defined on the complement C\Eof a dense polar set E ⊂ C. In order to illustrate results of Chapter 5 a secondexample, which appeared in [41], is elaborated in subsection 7.2.2. Moreover, acollection of open problems is presented in Section 7.3.

7.1 h-Hausdorff Measure

Let h be a real valued, increasing function on the interval [0, 1) with limr→0 h(r) =h(0) = 0. Such a function is sometimes called a dimension function, see [53]. Theh-Hausdorff measure of a set E ⊂ C is defined as follows:

mh(E) = limδ→0

(

inf∑

k

h(rk)

)

, (7.1.1)

where the infimum is taken over all coverings of E by balls Bk with radii rk notexceeding δ. Note that if h(r) = rs, the mrs is the usual s-dimensional Hausdorffmeasure.

We will use the following known result. It is due to Erdos and Gillis [52]. Asimpler proof of it was given by Carleson [20].

Theorem 7.1.1. Let E ⊂ C. If E has finite h-Hausdorff measure with respect tothe function h(r) = (log(1/r))−1, then it is polar.

91

92 Chapter 7. Examples and Open Questions

7.2 Borel Series

7.2.1 Example 1

Let aj∞j=1 be a dense sequence in C with the property that |aj | < j, and let0 < cj < 1, j = 1.2.... Suppose that

1

ej2 ≤ cj < 1, (7.2.1)

and∞∑

j=n

−1

log cj< +∞. (7.2.2)

The inequality (7.2.1) guarantees the existence of a natural number m0 ≥ 4such that

1

nm0log(Πn

j=1cj) → 0, as n→ ∞. (7.2.3)

Let E =⋂

m

⋃

j≥m B(aj , cj). By Baire’s theorem, E is a dense subset of C.Moreover, E is polar in view of (7.2.2) and Theorem 7.1.1.

Denote by D the complement of E:

D =⋃

m

(C\⋃

j≥m

B(aj , cj))def=⋃

m

Fm.

Observe that (Fm) is an increasing sequence of closed sets.Let

f(z) =∞∑

j=1

ck(j)+1j

(z − aj), (7.2.4)

where (k(j)) is a sequence that will be determined later on.Let

Γf (C\(E ∪ aj∞j=1)) = (z, f(z)) : z ∈ C\(E ∪ aj

∞j=1)

Proposition 7.2.1. There exist a polar set E ⊇ E ∪ aj∞j=1 and a plurisub-

harmonic function ψ ∈ PSH(C2) such that

ψ = −∞ = Γf (C\(E ∪ aj∞j=1) ∪ (E × C). (7.2.5)

Proof. Let rn(z) =∑n

j=1

ck(j)+1j

(z−aj), and let qn(z) = (z − a1)...(z − an). Put

hn(z, w) :=1

nm0log |(w − rn(z))qn(z)| (7.2.6)

Then hn is a continuous plurisubharmonic function in C2.For any ν ≥ 2 denote by Dν the disk with center 0 and radius ν, and put

Dν = Dν ∩ Fν .

7.2. Borel Series 93

For z ∈ Fν\aj : j = 1, ..., ν − 1 we have, for n ≥ ν,

|f(z) − rn(z)| = |∞∑

j=n+1

ck(j)+1j

(z − aj)| ≤

∞∑

j=n+1

ck(j)j . (7.2.7)

Now we choose k(j) such that

∞∑

j=n+1

nnm0ck(j)j ≤ n, for all n ∈ N. (7.2.8)

For z ∈ Dν\aj : j = 1, ..., ν − 1, we have, using (7.2.7) and (7.2.8)

|f(z) − rn(z)|1/nm0≤ (

1

nnm0

∞∑

j=n+1

nnm0ck(j)j )1/nm0

≤1

nn1/nm0

= (1

n)1−

1nm0 .

Therefore||f(z) − rn(z)||

1/nm0

Fν→ 0. (7.2.9)

For z ∈ Dν\aj : j = 1, ..., ν − 1 and n ∈ N we have

hn(z, f(z)) :=1

nm0log |(f(z) − rn(z))qn(z)|

=1

nm0log |(f(z) − rn(z))| +

1

nm0log |qn(z)|

≤ log ||f − rn||1/nm0

Dν+

1

nm0log |(z − a1)...(z − an)|

≤ log ||f − rn||1/nm0

Dν+

1

nm0log(ν + n)n. (7.2.10)

Inequality (7.2.10) holds because |aj | < j, and |z| < ν. Hence, for any ν ∈ N thereexists n1(ν) such that, for all n ≥ n1(ν), we have

hn(z, f(z)) ≤ −ν, z ∈ Dν\aj : j = 1, ..., ν − 1. (7.2.11)

Fix ν ∈ N. For z ∈ Dν\aj : j = 1, ..., ν − 1 and w ∈ Dν , with |w − f(z)| > 1/νand n ≥ n1(ν) we have, using (7.2.11)

hn(z, w) :=1

nm0log |(w − f(z))qn(z) + (f(z) − rn(z))qn(z)|

≥1

nm0log |

1

ν|qn(z)| − exp(−νnm0)|

≥1

nm0log(Πn

j=1cj) −1

nm0log(ν) − 1. (7.2.12)


In view of (7.2.3), there exists n2(ν) ≥ n1(ν) such that for n ≥ n2(ν) we have

hn(z, w) ≥ − log(ν) − 2. (7.2.13)

Let us estimate hn on Dν × Dν . Let ν + 1 > R > ν such that |z| = R ⊂ Fν .

Since ||f − rn||1/nm0

z∈C: |z|=R → 0, for sufficiently big n ∈ N we have

max|z|≤R

|pn(z)| = max|z|=R

|pn(z)| ≤ CR max|z|=R

|qn(z)|,

where CR = max|z|=R |f(z)| + 1. Hence there exists n3(ν) ≥ n2(ν) such that, forn ≥ n3(ν), (z, w) ∈ Dν × Dν , we have

hn(z, w) :=1

nm0log |wqn(z) − pn(z)|

≤1

nm0log(ν(ν + n)n + CR(ν + n)n)

≤ log(ν + 2) (7.2.14)

For any ν, fix an n(ν) ≥ n3(ν). Now, we consider the plurisubharmonic functions

vν(z, w) = maxhν(n)(z, w) − log(ν + 2),−ν − log(ν + 2)

As vν(z, w) is negative on Dν × Dν , the series

v(z, w) :=

∞∑

ν=2

1

ν2vν(z, w) (7.2.15)

represents a plurisubharmonic function in C2. It is not identically −∞, becauseof (7.2.13), while (7.2.11) shows that it is −∞ on the graph of f over D\aj :

j = 1, .... Let ϕ ∈ SH(C) such that the set ϕ = −∞, which we denote by E,contains E ∪ aj∞j=1. Observe now that the function ψ(z, w) = v(z, w) + ϕ(z)satisfies all our conditions. This ends the proof.

It is possible to obtain more precise information on the pluripolar hull of thegraph of f , by making the following extra assumption on the sequence aj∞j=1:

Suppose that for every j, aj belongs to infinitely many balls B(ak, ck), k ≥ 1.

Under the above assumption it is clear that aj∞j=1 is a subset of E. Since Eis a polar and a Gδ , There exists a function ϕ ∈ SH(C), such that E = ϕ = −∞.Hence the above propositions can be reformulated as follows.

Corollary 7.2.2. There exists a plurisubharmonic function ψ ∈ PSH(C2) suchthat

(z, w) ∈ C2 : ψ(z, w) = −∞ = Γf (C\E) ∪ (E × C). (7.2.16)

7.2. Borel Series 95

7.2.2 Example 2

Here we give an other example in the spirit of Borel to which the results fromChapter 5 apply (cf, [41]). It consists of a finely holomorphic function on an F-domain, which is a dense subset of C with empty Euclidean interior. Our point isto show that the study of quite natural series in connection with pluripolarity isfruitfully done in the framework of fine holomorphy.

Let aj∞j=1 be a dense sequence in C with the property that |aj | < j. Let

rj = 2−j . Then ∪∞j=1B(aj , rj) has finite area, and its circular projection z 7→ |z|

has finite length. Next, define subharmonic functions gj(z) = log |z− aj | − 3j andun by

un(z) =

∞∑

j=n

j−3gj(z). (7.2.17)

The terms in the sum of (7.2.17) are subharmonic and they are negative for |z| <k as soon as j > k. Hence un represents a subharmonic function. Let D =(∪nun > −10) \ a1, a2, . . .. We claim that D = u1 > −∞. Indeed, letz0 ∈ ∪nun > −∞\a1, a2, . . .. Then there exists a natural number k such that|z0| < k and uk > −∞. Since all the terms of the series uk(z0) are negative, asuitable tail, say uN (z0), will be very close to 0. In other words, z0 ∈ uN > −10.Hence z0 ∈ D and consequently D = ∪nun > −∞ \ a1, a2, . . .. Therefore,

C\D = ∩∞n=1uk = −∞ ∪ a1, a2, . . ..

Since uk1 = −∞\a1, a2, . . . = uk2 = −∞\a1, a2, . . . for any natural num-bers k1 and k2, we conclude that

C\D = u1 = −∞ ∪ a1, a2, . . . = u1 = −∞.

This proves the claim. In particular, D is, by Theorem 2.2.29, an F-domain.For every j there exists 0 < cj < 1 such that if |z − aj | < cj , then for n ≤ j,

un(z) < −11. Indeed,∑

k>j k−3gk(z) < 0, while

j−1∑

k=n

k−3gk(z) < log j

j−1∑

k=n

k−3 < 10 log j.

So it suffices to take cj = j−11j3

.Next we define a function on D by

f(z) =

∞∑

j=1

cj2j(z − aj)

, (7.2.18)

We claim that the function f is finely holomorphic on D. Indeed, let z0 ∈ D. Forevery m a suitable tail of the series of f in (7.2.18) is uniformly convergent on thecompact set K = |z| ≤ 2|z0| \ ∪j≥mB(aj , cj). Now if z0 ∈ D, then z0 belongs


to the finely open set um > −10 for some m. Hence, for all j ≥ m we have|z0 − aj | > cj , and K is a F-neighborhood of z0.

Application of Corollary 5.3.2 gives us that the graph of Γf (D) of f over D is apluripolar set. The theorem also shows that for a set of positive capacity E ⊂ D,e.g., a circle in u1 > −10,

Γf (D) ⊂ (Γf (E))∗C2 .

Even for this example there are many questions left. We have no descriptionof the maximal domain D0 to which f extends as a finely holomorphic function,and we don’t know if Γf (D0) = (Γf (E))

∗C2 , as one may expect in view of [40].

7.3 Unsolved Problems

In this section we collect some problems that we could not solve.In view of Example 7.2.2 it seems quite natural to ask the followingProblem 1. Let E be a polar dense subset of C, and let f be a finely holo-

morphic functions in the fine domain C\E. Let Γf (C\E) be the graph of f overC\E, and let Γ∗

f (C\E) be its pluripolar hull.Is it true that Γ∗

f (C\E) is contained in Γf (C\E) ∪ (E × C)?.Problem 2. Let f : U −→ C be a finely continuous function on a finely open

subset U of C. Suppose that the graph of f over U is a pluripolar subset of C2.Is it true that f is finely holomorphic in U?

The answer to this problem in the case of a usual open set U was given byShcherbina [124]. In the present situation, with f is a C2 function, a positiveanswer was given by Edlund in his thesis. Very recently, Edigarian and Wiegerinck[37] showed that the answer is still affirmative for C1 functions.

We have seen in Proposition 5.5.1 that if a pluripolar set E hits a finely analyticcurve in a “big” set, then the pluripolar hull of E must contain the whole curve.Thus, in order to understand the phenomenon of propagation of pluripolar sets, itseems necessary to study in more details the concept of fine analytic continuation.In particular we are led to the following two problems.

Problem 3. Let f : U −→ C be a finely holomorphic functions on a finelyopen subset U of C. The classical process of analytic continuation and the Weier-strass’s concept of the maximal analytic function can be carried over to the presentsituation, see Definition 5.2.3. The classical Poincare-Voltera theorem, see [110],states that the maximal analytic function of any given holomorphic function cannot have more that countably many values. See also [130] for the history of thisresult.

The question is: is there an analogue to the classical Poincare-Voltera theoremfor f?. In other words, if F is the “maximal” fine analytic functions of f , is ittrue that F has at most countably many values at each point, or at least at eachpoint outside some polar set?.

Of course, Poincare’s proof relies decisively on the fact that the usual topologyhas a countable base. Since the fine topology does not have this property, the

7.3. Unsolved Problems 97

Poincare’s argument can not be directly carried over. However, it seems that thequasi-Lindelof property, cf. Theorem 2.2.20, might serve as a replacement for thecountable base argument.

Problem 4. In [45] Edlund proved that for any non-empty closed subsetK ⊂ C there exists a continuous function f(z) on K such that its graph Γf (K)over K is complete pluripolar in C2. The function f in Edlund’s theorem iscertainly finely holomorphic in the F-interior K ′ of K (if K ′ 6= ∅). Accordingto Theorem 5.4.1, f has no fine analytic continuation outside K ′. This meansthat K ′ is an F-domain of fine holomorphy. One can therefore ask the followingquestionLet U ⊆ C be an F-domain. Is it true that U is an F-domain of fine holomorphy?.

In analogy with the theory of finely subharmonic functions one can pose thefollowing problem, see Theorem 2.3.19.

Problem 5. Let f : U −→ [−∞,+∞[ be F-plurisubharmonic in an F-domainU ⊂ Cn. Suppose that every point z ∈ U has a compact F-neighborhood Kz ⊂ Usuch that f |Kz is continuous in the usual sense.Is it true that f |Kz is the uniform limit of a sequence of usual plurisubharmonicfunctions defined in Euclidean neighborhoods of K.

Problem 6. In connections to Problem 5 it seems also interesting to askwhether Theorem 4.3.3 holds without the exceptional pluripolar set E.

Bibliography

[1] Ahlfors, L. V., Untersuchungen zur Theorie der konformen Abbildung und derganzen Funktionen, Acta. Soc. Sci. Fenn., Nova Ser. I, no. 9 (1930).

[2] Armitage, D. H. and Gardiner, S. J.: Classical potential theory. SpringerMonographs in Mathematics. Springer-Verlag London, Ltd., London, 2001.

[3] Barton, A.: Condition on Harmonic Measure Distribution Functions of PlanarDomains, Thesis, Harvey Mudd College Mathematics 2003.

[4] Bedford, E. and Taylor, B. A: A new capacity for plurisubharmonic functions,Acta Math, 149, 1–40 (1982).

[5] Bedford, E. and Taylor, B. A.: Fine topology, Silov boundary and (ddc)n, J.Funct. Anal. 72 (1987), 225–251.

[6] Bedford, B.: Survey of pluripotential theory, Several complex variables: Pro-ceedings of the Mittag-Leffler Inst. 1987-1988 (J-E. Fornæss, ed), Math. Notes,38, Princeton University Press, Princeton, NJ, 1993.

[7] Betsakos, D.: Geometric theorems and problems for harmonic measure, RockyMountain J. Math., 31, no. 3,(2001), 773–795 .

[8] Beurling, A.: Etudes sur un problem de majoration. Thesis. Upsala. 1933.

[9] Boboc, N., Constantinescu, C., Cornea, A.: Axiomatic theory of harmonicfunctions. Balayage. Ann. Inst. Fourier. (Grenoble) 15 (1965), fasc. 2, 37–70.

[10] Borel, E.: Sur quelques points de la theorie des fonctions. Annales scien-tifiques de l’cole Normale Suprieure Sr. 3, 12 (1895), 9–55.

[11] Borel, E.: Les fonctions monognes non analytiques. Bulletin de la Socit Math-matique de France, 40 (1912), 205–219.

[12] Borel, E.: Sur les fonctions monogenes uniformes d’une variable complexe.Gauthier Villars, Paris. 1917.

[13] Brelot, M.: Points irreguliers et transformations continues en theorie du po-tentiel. (French) J. Math. Pures Appl. (9) 19, (1940), 319–337.

99

100 Bibliography

[14] Brelot, M.: Sur les ensembles effiles. Bull. Sci. Math. (2) 68, (1944). 12–36.

[15] Brelot, M.: Minorantes sous-harmoniques, extremales et capacites. J. Math.Pures Appl. (9) 24, (1945). 1–32.

[16] Brelot, M.: On Topology and Boundaries in Potential Theory, Lecture Notesin Mathematics, 175, Springer-Verlag, Berlin, 1971.

[17] Brelot, M.: Elements de la Theorie Classique du Potentiel. Centre de Docu-mentation Universitaire, Paris, 1959.

[18] Bremermann, H. J.: On the conjecture of the equivalence of the plurisub-harmonic functions and the Hartogs functions Math. Ann. 131 (1956), 76–86.

[19] Bremermann, H. J.: Die Holomorphiehullen der Tubenund Halbtubengebiete,Math. Ann., 127 (1954), 406–423.

[20] Carleson, L.: On the connection between Hausdorff measures and capacity.Ark. Mat. 3 (1958), 403–406.

[21] Cegrell, U, Poletsky, E.: A counterexample to the “fine” problem in pluripo-tential theory. Proc. Amer. Math. Soc. 123 (1995), no. 12, 3677–3679.

[22] Cegrell, U.: Thin sets in Cn, Univ. Iagel. Acta Math. No. 25 (1985), 115–120.

[23] Choquet, G.: Lectures on Analsis, Vol. I. New York-Amsterdam: W. A.Benjamin 1969.

[24] Choquet, G.: Sur les Gα de capacite nulle. Ann. Inst. Fourier. Grenoble 9(1959), 103–109.

[25] Coman. D., Levenberg, N., Polytsky, E. A.: Smooth submanifolds intersectingany analytic curve in a discrete set, Math. Ann., 332 (2005), 55–65.

[26] Constantinescu, C.: Some properties of the balayage of measures on a har-monic space. Ann. Inst. Fourier (Grenoble) 17 (1967) fasc. 1, 273–293.

[27] Constantinescu, C., Cornea, A: Potential Theory on Harmonic Spaces,Springer-Verlag, Berlin/Heidelberg/ New York, 1972.

[28] Davie, A., Øksendal, B.: Analytic capacity and differentiability properties offinely harmonic functions, Acta Mathematica Volume 149 (1982), 127–152.

[29] Debiard, A., Gaveau, B.: Potential fin et algebres de fonctions analytiques, I,J. Functional Anal. 16 (1974), 289–304, II, 17 (1974), 296–310.

[30] Debiard, A., Gaveau, B.: Differentiabilite des fonctions finement har-moniques. Inventiones Math. 29 (1975), 111–123.

[31] Denjoy, A.: Sur les fonctions derivees sommables. Bull. Soc. Math. France,43 (1915), 161–248.

Bibliography 101

[32] Deny, J., Lions, J. L.: Les espaces du type de Beppo Levi,Ann. Inst. Fourier,5, (1953-1954), 305–370.

[33] Diederich, K., Fornaess, J. E.: A smooth curve in C2 which is not a pluripolarset. Duke. Math. J., 49 (4) (1982), 931–936.

[34] Doob, J. L.: Semimartingales and subharmonic functions. Trans. Amer.Math. Soc. 77, (1954). 86–121.

[35] Doob, J. L.: Applications to analysis of a topological definition of smallnessof a set. Bull. Amer. Math. Soc. 72 (1966), 579–600.

[36] Doob, J. L.: ”Classical Potential Theory and Its Probabilistic Counterpart,” Springer-Verlag, Berlin, 1984.

[37] Edigarian, A., Wiegerinck, J.: Shcherbina’s Theorem for finely holomorphicfunctions. arXiv:0810.4878v1.

[38] Edigarian, A., Wiegerinck, J.: The pluripolar hull of the graph of a holo-morphic function with polar singularities, Indiana Univ. Math. J., 52 (2003),1663–1680.

[39] Edigarian, A., Wiegerinck, J.: Graphs that are not complete pluripolar, Proc.Amer. Math. Soc. 131 (2003), 2459–2465.

[40] Edigarian, A., Wiegerinck, J.: Determination of the pluripolar hull of graphsof certain holomorphic functions, Ann. Inst. Fourier. Grenoble, 54 (2004),no. 6, 2085–2104.

[41] Edigarian A., El Marzguioui, S., Wiegerinck, J.,: The image of a finely holo-morphic map is pluripolar, Preprint. arXiv: math/0701136.

[42] Edigarian A., The Pluripolar Hull and the Fine Topology, Bull. Pol. Acad.Sci. Math. 53 (2005), no. 3, 285–290.

[43] Edlund, T.: Pluripolar sets and pluripolar hulls, Uppsala Dissertations inMathematics 41 2005.

[44] Edlund, T. and Joricke, B.: The pluripolar hull of a graph and fine analyticcontinuation, Ark. Mat.44 (2006), no. 1, 39–60.

[45] Edlund, T.: Complete pluripolar curves and graphs, Ann. Polon. Mat. 84(2004), 75-86.

[46] Edlund, T., El Marzguioui, S.: Pluripolar hulls and fine analytic structure,to appear in Indagationes Mathematicae.

[47] El Kadiri, M.: Fonctions finement plurisousharmoniques et topologie plu-rifine. Rend. Accad. Naz. Sci. XLMem. Mat. Appl. (5) 27, 77–88 (2003).

102 Bibliography

[48] El Marzguioui, S. Wiegerinck, J.: The pluri-fine topology is locally connected.Potential Anal 25 (2006), no. 3, 283–288.

[49] El Marzguioui, S., Wiegerinck, J.: Connectedness in the plurifine topology, toappear in Proceedings of the Conference on Functional Analysis and ComplexAnalysis Edited by: A. Aytuna, R. Meise, T. Terzioglu, and D. Vogt.

[50] El Marzguioui, S., Wiegerinck, J.: Continuity Properties of Finely Plurisub-harmonic Functions and Pluripolarity, preprint 2008.

[51] El Mir, H.: Sur le prolongement des courants positifs fermes. Acta. Math.153 (1984), no. 1-2, 1–45.

[52] Erdos, P., Gillis, J.: Note on the transfinite diameter, J. London Math. Soc.12 (1937), 185–192.

[53] Falconer, K.: Fractal geometry. Mathematical foundations and applications,1990.

[54] Favorov, S. Yu., On a problem of V. P. Vladimirov, Functional. Anal. Appl.12 (1978), 236–237.

[55] Fuglede, B.: Connexion en topologie fine et balayage des mesures, Ann. Inst.Fourier Grenoble 21 (1971), no. 3, 227–244.

[56] Fuglede, B.: Finely harmonic functions, Springer Lecture Notes in Mathemat-ics, 289, Berlin-Heidelberg-New York, 1972.

[57] Fuglede, B.: Fonctions harmoniques et fonctions finement harmoniques, Ann.Inst. Fourier Grenoble 24 (1974), no. 4, 77–91.

[58] Fuglede, B.: Asymptotic paths for subharmonic functions, Math. Ann, 213(1975), 261–274.

[59] Fuglede, B.: Sur la fonction de Green pour un domaine fin. Ann. Inst. Fourier,25 no. 3-4 (1975), 201–206.

[60] Fuglede, B.: Finely harmonic mappings and finely holomorphic functions,Ann. Acad. Sci. Fennicæ, 2 (1976), 113–127.

[61] Fuglede, B.: Sur les fonctions finement holomorphes, Ann. Inst. Fourier. 31(1981), no. 4, 57–88.

[62] Fuglede, B.: Fine topology and finely holomorphic functions, ”Proc. 18thScand. Congr. Math. Aarhus 1980”, Birkhauser, (1981), pp. 22–38.

[63] Fuglede, B.: Localisation in fine potential theory and uniform approximationby subharmonic functions, J. Functional Anal. 49 (1982), 57–72.

Bibliography 103

[64] Fuglede, B.: Fonctions BLD et fonctions finement surharmoniques, ”Seminaire de Therie du Potentiel, Paris, No. 6,” Lecture Notes in Mathe-matics No. 906, Springer, Berlin/Heidelberg/New York, 1982, pp. 126–157.

[65] Fuglede, B.: Fonctions finement holomorphes de plusieurs variables - un essai.Seminaire d‘Analyse P. Lelong P. Dolbeault - H. Skoda, Springer - Verlag,Lecture Notes in Math. 1198 (1986), 133–145.

[66] Fuglede, B.: Finely holomorphic functions- a survey, Revue Roumaine Math.Pures Appl. 33 (1988), 283–295.

[67] Fuglede, B.: Asymptotic paths for subharmonic functions and polygonal con-nectedness of fine domains Seminar on Potential Theory, Paris, No. 5, pp.97–116, Lecture Notes in Math., 814, Springer, Berlin, 1980.

[68] Fuglede, B.: On the mean value property of finely harmonic and finely hy-perharmonic functions, Aequationes Math. 39 (1990), 198–203.

[69] Fuglede, B.: Approximation by solutions of partial differential equations, 195-201. Kluwer Academic Publishers Group, Dordrecht, 1992. MR1168702.

[70] Fuglede, B.: Personal communication.

[71] Gamelin, T. W. and Lyons, T. J.: Jensen measures for R(K), J. LondonMath. Soc. 27 (1983), 317–330.

[72] Gauthier, P. M., Ladouceur, L.: Uniform approximation and fine potentialtheory, J. Approx. Theory 72.2 (1993), 138-140.

[73] Goffman, C., Waterman, D.: Approximately continuous transformations.Proc. Amer. Math. Soc, 12 (1961), 116–121.

[74] Hartogs, F.: Zur Theorie der analytischen Funktionen mehrerer unabhangigerVeranderlichen, insbesondere uber Darstellung derselben durch Reihen,welche nach Potenzen einer Veranderlichen fortschreiten. Math. Ann. 62(1906), no. 1, 1–88.

[75] Herve, R. M.: Recherches axiomatiques sur la theorie des fonctions surhar-moniques et du potentiel. Ann. Inst. Fourier. (Grenoble) 12 (1962), 415–571.

[76] Helms, L.L.: Introduction to potential theory, Pure and Applied Mathematics,Vol XXII. Wiley-Interscience, New York, 1969.

[77] Hormander, L.: Notions of convexity, Progress in Mathematics, Birkhuauser,Boston, 1994.

[78] Josefson, B.: On the equivalence between locally polar and globally polar setsfor plurisubharmonic functions on (Cn) , Ark. Math. 16 (1978), 109–115.

104 Bibliography

[79] Kiselman, C. O.: Plurisubharmonic functions and potential theory in sev-eral complex variables. Development of mathematics 1950–2000, 655–714,Birkhuauser, Basel, 2000.

[80] Klimek, M.: Pluripotential Theory, London Mathematical Society Mono-graphs, 6, Clarendon Press, Oxford, 1991.

[81] Labutin, D. A.: Pluripolarity of sets with small Hausdorff measure.Manuscripta Math. 102 (2000), no. 2, 163–167.

[82] Landkof, N. S.: Foundations of modern potential theory. Springer-Verlag,New York-Heidelberg, 1972.

[83] Lelong, P.: Sur quelques problemes de la theorie des fonctions de deux vari-ables complexes, Ann. Sci. Ecole Norm. Sup. 3 58 (1941), 83–177.

[84] Lelong, P.: Definition des fonctions plurisousharmoniques, C. R. Acad. Sci.Paris 215, (1942), 398–400.

[85] Lelong, P.: Les fonctions plurisousharmoniques, Ann. Sci. Ecole Norm. Sup.62 (1945), 301–328.

[86] Lelong, P.: Ensembles singuliers impropres des fonctions plurisoushar-moniques. (French) J. Math. Pures Appl. 9 36 (1957), 263–303.

[87] Lelong, P.: Fonctions plurisousharmoniques et fonctions analytiques de vari-ables reelles, Ann. Inst. Fourier. 11 (1961), 515–562.

[88] Lelong, P.: Fonctions entieres de type exponentiel dans Cn Ann. Inst. Fourier.16 (1966), no. 2, 269-318.

[89] Lelong, P.: Fonctions plurisousharmoniques et formes differentielles positives.Dunod, Paris (1968).

[90] Lelong, P., Gruman, L.: Entire functions of several complex variables.Springer-Verlag, Berlin, 282, 1986.

[91] Levenberg, N.: On an example of Wermer, Ark. Mat. 26 (1988), 155–163.

[92] Levenberg, N., Slodkowski, Z.: Pseudoconcave pluripolar sets in C2, Mat.Ann. 308 (1998), 429–443.

[93] Levenberg, N., Poletsky, E.: Pluripolar hulls, Mich. Math. J. 46 (1999), 151–162.

[94] Levenberg, N: Math Reviews, MR 2255349.

[95] Lukes, J., Maly J.: Fine Hyperharmonicity Without Axiom D, Math. Ann.261 (1982), 299-306.

Bibliography 105

[96] Lukes, J., Maly, J., Zajcek, L.: Fine topology methods in real analysis andpotential theory Lecture Notes in Mathematics, No. 1189, Springer-Verlag,Berlin, (1986).

[97] Lyons, T. J.: Finely holomorphic functions, J. Functional Anal. 37 (1980),1–18.

[98] Lyons, T. J.: A theorem in fine potential theory and applicatios to finelyholomorphic functions, J. Functional Anal. 37 (1980), 19–26.

[99] Lyons, T. J.: Cones of lower semicontinuous functions and a characterisationof finely hyperharmonic functions. Math. Ann. 261 (1982), no. 3, 293–297.

[100] Lyons, T. J.: Finely Harmonic Functions need not be Quasi-Analytic Bull.London Math. Soc. 16 (1984), 413–415.

[101] Lyons, T. J.: An application of fine potential theory to prove a Phragmen-Lindelof theorem. Ann. Inst. Fourier (Grenoble) 34 (1984), no. 2, 63–66.

[102] Mastrangelo-Dehen, M., Dehen, D.: Etudes des fonctions finement har-moniques sur des ouverts fins. C. R. Acad. Sci., Paris, Ser. A 275 (1972),361–364.

[103] Morosawa, S., Nishimura, Y., Taniguchi, M., Ueda, T.: Holomorphic dy-namics. Cambridge Studies in Advanced Mathematics, 66. 2000.

[104] Nevanlinna, R.: Uber eine Minimumaufgabe in der Theorie der KonformenAbbildung, Nachr. Ges. Wiss. Gottingen I, 37 (1933), 103–115.

[105] Nguyen-Xuan-Loc., Watanabe, T.: Characterization of fine domains for acertain class of Markov processes, Z. Wahrscheinlichkeitstheorie u. verw. Geb.21, (1972), 167–178.

[106] Nguyen-Xuan-Loc.: Sur la theorie des fonctions finement holomorphes. Bull.Sci. Math., 102 (1978), 271–308.

[107] Nguyen-Xuan-Loc.: Singularities of locally analytic processes (with applica-tions in the study of finely holomorphic and finely harmonic functions). In:Potential Theory, Copenhagen, (1979), 267–288. Springer Lecture Notes inMathematics, 787. Berlin-Heidelberg-New York, (1980).

[108] Oka, K.: Sur les fonctions analytiques de plusieurs variables. VI. Domainespseudoconvexes. Tohoku Math. J. bf 49, (1942). 15–52.

[109] Pflug, P., Nguyen, V. A.: Extension theorems of Sakai type for separatelyholomorphic and meromorphic functions., Ann. Polon. Math 82 (2003), no.2,171–187.

[110] Poincare, H.: Sur une propriete des fonctions analytiques, OEuvres de HenriPoincare, Vol 4, Page 11-13.

106 Bibliography

[111] Poincare, H.: Sur les Equations aux Derivees Partielles de la PhysiqueMathematique Amer. J. Math., 12 : 3 (1890), 211–294.

[112] Poincare, H.: Sur les Fonctions a Espaces Lacunaires. American Journal ofMathematics, Vol. 14, No. 3, (1892), 201–221.

[113] Poletsky, E.: Analytic geometry on compacta in Cn, Math. Z. 222 (1996),407–424.

[114] Poletsky, E.: Holomorphic currents, Indiana Univ. Math. J. 42 (1993), 85-144.

[115] Pyrih, P.: Finely holomorphic functions and quasi-analytic classes. PotentialAnal. 3, no. 3 (1994), 273–281.

[116] Ransford, T. J.: Open mapping, inversion and implicit function theoremsfor analytic multivalued functions. Proc. London Math. Soc. 49 (1984), no. 3,537–562.

[117] Ransford, Th.: Potential Theory in the complex Plane, Cambridge Univer-sity Press, (1994).

[118] Remmert, R.: Classical topics in complex function theory. Graduate Textsin Mathematics, 172. Springer-Verlag, New York, 1998.

[119] Riesz, F.: Sur les fonctions subharmoniques et leur rapport a la theorie dupotentiel. Acta. Math., 48 (1926), 329–343.

[120] Rudin, W.: Real and Complex Analysis, McGraw-Hill Book Company, NewYork, 1966.

[121] Sadullaev, A.: P -regularity of sets in Cn. Lecture Notes in Math., Springer-Verlag. Berlin, 798 (1979), 402–408.

[122] Sadullaev, A.: Plurisubharmonic measures and capacities on complex man-ifolds, Russian Math. Surveys, 36 (1981), 61–119.

[123] Sadullaev, A.: A criterion for fast rational approximation in Cn. (Russian)Mat. Sb. (N.S.) 125 (167) (1984), no. 2, 269–279.

[124] Shcherbina, N. V.: Pluripolar graphs are holomorphic, Acta. Math., 194(2005), 203–216.

[125] Siciak, J.: Pluripolar sets and pseudocontinuation, in: Complex Analysisand Dynamical Systems II (Nahariya, 2003), Contemp. Math. 382, Amer.Math. Soc., Providence, RI, 2005, 385–394. 22 (2005), 195–206.

[126] Siu, Y. T.: Extension of meromorphic maps into Kahler manifolds, Annals.of Math, 102, (1975), 421–462.

Bibliography 107

[127] Slodkowski, Z., Analytic set-valued functions and spectra, Math. Ann. 256(1981), 363-386.

[128] Sylvain, R.: Extreme Jensen measures, Ark. Mat., 46 (2008), 153–182.

[129] Thorbiornson, J.: A counterexample to the strong subadditivity of extremalplurisubharmonic functions, Monatshefte fur Mathematik. 105 (1988), 245–248.

[130] Ullrich, P.: The Poincare-Volterra theorem: from hyperelliptic integrals tomanifolds with countable topology. Arch. Hist. Exact Sci. 54 (2000), no. 5,375–402.

[131] Vladimirov, V. S.: Methods of the Theory of Functions of Several ComplexVariables. (1966).

[132] Wermer, J.: Polynomially convex hulls and analyticity , Ark. Math. 20(1982), 129–135.

[133] Wiegerinck, J.: The pluripolar hull of w = exp−1/z, Ark. Mat. 38 (2000),201–208.

[134] Wiegerinck, J.: Graphs of holomorphic functions with isolated singularitiesare complete pluripolar, Michigan Math. J., 47 (2000), 191–197.

[135] Wiegerinck, J.: Pluripolar sets: hulls and completeness. Actes des Rencon-tres d’Analyse Complexe (Poitiers-Futuroscope, 1999), 209–219.

[136] Wiegerinck, J.: Fine function theory. (Dutch) Nieuw Arch. Wiskd. (5) 8(2007), no. 2, 82–88.

[137] Winkler, J.: A Uniqueness Theorem For Monogenic Functions. Ann. Acad.Sci. Fennicae Ser. A. I. Math. 18 (1993), 105–116.

[138] Zaremba, S.: Sur le principe de Dirichlet, Acta Math. 34 (1911), no.1, 293–316.

[139] Zeriahi, A.: Ensembles pluripolaires exceptionnels pour la croissance par-tielle des fonctions holomorphes , Ann. Polon. Math. 50 (1989), 81–91.

[140] Zwonek, W.: A note on pluripolar hulls of graphs of Blaschke products,Potential Analysis, 22 (2005), 195–206.

Index

A′

, 28A, 28arcwise connected, 34approximately continuous function, 33

BϕC, 28

BϕB(a,r), 29

BϕΩ, 51

b(A), 28Baire, 29, 33, 49, 92balayage, 13Bedford, 49Beurling, 54Borel’s Monogenic Functions, 45Borel series, 92Brelot property, 40Brelot, 13Bremermann’s theorem, 15

A, 28Carleman-Milloux, 54Cartan, 13, 50completely regular, 30, 49convex function, 15complete pluripolar set, 17, 82, 88

∂fA, 28d(z, A), 54density topology, 33Dirichlet problem, 25

E∗Ω, 18

E∗C2 , 76, 88

Edigarian, 19, 75, 96Edlund, 70, 72, 97εE

z , 36

F-PSH(U), 62finely analytic curve, 76, 80Fine Analytic Structure, 79, 85F-holomorphic functions, 71finely harmonic function, 38, 42finely holomorphic function, 41finely hypoharmonic, 37fine topology, 27, 28Finely Plurisubharmonic Functions,

62f- lim sup, 39F-pluripolar set, 62, 66Fine Pluripotential Theory, 21finely subharmonic function, 35, 37,

52, 73, 76, 82FSH(U), 37Fuglede, 21, 57, 81

Gauthier, 21gluing lemma, 59Green set, 25

H(K), 71Harmonic measure, 54harmonic function, 25Hessian matrix, 14

=z, 54i(A), 28irregular boundary point, 26

Joricke, 70, 80

Levenberg, 18, 19, 70, 80local connectedness, 21, 34, 51

maximum principle, 12

108

Index 109

mean-value inequality, 11, 12

Nevanlinna, 54Notations, 50

Open Problems, 88, 96Overview of the thesis, 20

path connected, 89pluripolar hull, 18, 20, 22, 69, 70, 74pluripolar sets, 15pluri-thin set, 17, 50pluri-fine topology, 20, 49Plurisubharmonic function, 13Poletsky, 18, 19, 70Poincare, 13, 46Poincare-Voltera, 96polar set, 12pseudoconvex, 88PSH(Ω), 14

quasi-Lindelof property, 33, 49, 51

<z, 54regular boundary point, 26regular set, 25RE

u ,REu , 35, 36

Shcherbina, 72, 96SH−(Ω), 35subharmonic function, 11, 55swept-out Measure, 35swepping out, 13

Taylor, 49thin set, 26, 32

upper semi-continuous, 13uniqueness theorem, 45

Wermer, 80, 81Wiegerinck, 19, 75, 81, 88, 96Wiener’s criterion, 32

Zeriahi, 18, 70

110 Index

Samenvatting

In dit proefschrift bestuderen we pluripolaire omhulsels en ontwikkelen we de the-orie van fijne plurisubharmonische functies. Deze onderzoeksthema’s lijken op heteerst gezicht weinig met elkaar te maken te hebben. Maar fijne plurisubharmoni-sche functies helpen om pluripolaire omhulsels beter te kunnen begrijpen.

Pas drie jaar geleden hebben Edlund and Joricke [44] ontdekt dat fijne func-tietheorie de natuurlijk verklaring geeft voor een ingewikkelde verschijnsel, datwordt vertoond door pluripolaire omhulsels van grafieken van holomorfe functies.Dit onverwachte verband, eigenlijke al impliciet in een eerder artikel van Edigarianen Wiegerinck [39], heeft ons gemotiveerd om het concept van fijne plurisubhar-monische functies te introduceren en te bestuderen. Deze functies zijn de analogavan plurisubharmonische functies in open verzamelingen ten opzichte van de plu-rifijne topologie (de grofste topologie die de plurisubharmonische functies continumaakt).

Het beginpunt van dit proefschrift is dus een uitvoerig onderzoek naar de eigen-schappen van de plurifijne topologie. In hoofdstuk 3 bewijzen we onder meer datdeze topologie lokaal samenhangend is. Dit beantwoordt positief een meer dantwintig jaar oude vraag van Fuglede [65]. Verder gaan we diep in op de structuurvan plurifijne open verzamelingen. We tonen aan dat deze verzamelingen, in tegen-stelling tot wat de gecompliceerde structuur van de plurifijne topologie zou doenvermoeden, een aangenaam karakter hebben, zie Proposition 3.4.9. Verrassend ge-noeg, berust de argumentatie op een klassieke ongelijkheid van A. Beurling and R.Nevanlinna. Met sortgelijke argumenten geven we een alternatief en conceptueelmakkelijker bewijs van de lokale samenhang van de plurifijne topologie.

In hoofdstuk 4 bestuderen we de lokaal eigenschappen van fijne plurisubhar-monische functies. We bewijzen dat ieder begrensde fijne plurisubharmonischefunctie lokaal te schrijven is als het verschil van twee gewone plurisubharmonischefuncties. Dit heeft tot gevolg dat fijne plurisubharmonische functies continu zijnten opzichte van de plurifijne topologie, dus niet alleen maar boven semi-continuzoals de definitie verlangt. Een andere belangrijk resultaat is dat de verzamelingwaarin deze functies −∞ worden pluripolair is. Als gevolg hiervan kunnen weprecieze uitspraken over pluripolair omhulsels doen.

Hoofdstuk 5 gaat over grafieken van fijne holomorfe functies. De belangrijkstestelling uit dit hoofdstuk is dat deze pluripolair zijn.

In hoofdstuk 6 construeren we een voorbeeld a la Wermer van een pluripo-

111

112 Index

lair verzameling waarvan de pluripolair omhulsel heel groot is, maar geen fijneanalytische structuur heeft.

Curriculum Vitae

De auteur werd geboren op 15 april 1976 in Al Hoceima (Marokko). Na het behalenvan zijn Baccalaureaat Wiskunde aan het El Badissi Lyceum te Al Hoceima, begonhij zijn Wiskunde en Natuurkunde studie aan de Universiteit Mohamed Premierte Oujda (Marokko). Vanaf 1996 besloot hij zich verder te concentreren op dewiskunde waarop hij in juni 1998 cum laude afstudeerde.

In oktober 1998 verhuisde hij naar Rabat (Marokko), waar hij complex analyseen pluripotentiaaltheorie studeerde aan de Universiteit Mohammed 5. Hij schreefzijn scriptie, getiteld Fonction de Green sur un espace de Stein parabolique etapplications, onder begeleiding van professor A. Zeriahi en B. Jennane. Hieropstudeerde hij af in oktober 2000, waarna hij begon met het onderzoek die deelswordt verricht aan Laboratoire Emile Picard van de Universiteit Paul Sabatier inToulouse onder begeleiding van professor Ahmed Zeriahi.

In 2003 verhuisde hij naar Rotterdam. Kort daarna ging hij werken als gas-tonderzoeker aan KDV instituut. Tegelijker tijd deed hij een intensieve cursusNederlandse taal. Na het behalen van het diploma Nederlands als tweede taal inaugustus 2004, begon hij in oktober zijn promotieonderzoek onder begeleiding vanprofessor Jan Wiegerinck.

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