differential forms orthogonal to holomorphic functions or forms, and their properties

11 11X CBORCTBA
M3.llA TE.JibCTBO «HAYKA» CM6MPCKOE OT.llEJIEHME
HOBOCHI>HPCK 1975
Translated from the Russian by R. R. Simha Translation edited by Lev J. Leifman
1980 Mathematics Subject Classification. Primary 32A25, 32F15; Secondary 32A35, 32A45.
ABSTRACT. This book is devoted to the description of exterior differential forms orthogonal to holomorphic forms of degree n - p, 0 OliO p OliO n (in particular, to holomorphic functions if p = n) with respect to integration over the boundary of a bounded domain D in C". The Martinelli Bochner-Koppelman formula, which is an integral representation of exterior differential forms, is given, and the characteristic properties of the trace of a holomorphic function on the boundary are studied. The question of representation and multiplication of distributions lying in GD'(R2"- 1) is discussed with the aid of a-closed forms of type (n, n- I) with harmonic coefficients.
Ubrary of Congress Cataloging in Publication Data
Aizenberg, Lev Abramovich, 1937- Differential fortnl orthogonal to holomorphie functions or forms, and their
properties. (Translations of mathematical monographs; v. 56) Translation of: Differentsial'nye formy, ortogonal'nye golomorfnym
funkfsnam iii formam, i ikh svoistva. Bibliography: p. Includes index. 1. Holomorphic functions. 2. Exterior forms. 3. Differential forms.
I. Dautov, Sh. A. (Shamil' Abdullovich) II. Title. III. Series. QA331.A46313 1983 515.9
ISBN 0-8218-4508-X ISSN 0065-9282
TABLE OF CONTENTS
Preface ix
Introduction 1
CHAPTER I. Integral representation of exterior differential forms and its immediate consequences 5
1. The Martinelli-Boehner-Koppelman formula 5 2. Theorems on the saltus of forms 14 3. Characterization of the trace of a holomorphie form on the
boundary of a domain 28 4. Some eases of the solvability of the d.-problem 31
CHAPTER IT. Forms orthogonal to holomorphie forms 37 5. Polynomials orthogonal to holomorphie functions 37 6. Forms orthogonal to holomorphic forms: the ease of strictly
pseudo-convex domains 45 7. The general ease 47 8. Converse theorems 49
CHAPTER ill. Properties of d-elosed forms of type {p, n- 1) 53 9. The theorems of Runge and Morera 53 ·
10 .. The first Cousin problem, separation of singularities, and domains of existence 57
11. Theorems of approximation on compact sets 60
v
vi CONTENTS
CHAPTER IV. Some applications 67 12. Generalization of the theorems of Hartogs and F. and M. Riesz 67 13. On the general form of integral representations of holomorphic
functions 71 14. Representation of distributions in D'(R2n-l) by a-closed ex-
terior differential forms of type ( n, n - 1) 76 15. Multiplication of distributions in V'(R2n-l) 82
Brief historical survey and open problems for Chapters I-IV 85
CHAPTER V. Integral properties characterizing a-closed differential forms and holomorphic functions 89 16. A characteristic property of a-closed forms and forms of class B 89 17. Holomorphy of continuous functions representable by the Martin
elli-Bochner integral; criteria for the holomorphy of integrals of the Martinelli-Bochner type 94
18. The traces of holomorphic functions on the Shilov boundary of a circular domain 107
19. Computation of an integral of Martinelli-Bochner type for the case of the ball 111
20. Differential boundary conditions for the holomorphy of functions 116
CHAPTER VI. Forms orthogonal to holomorphic forms. Weighted formula for solving the a-equation, and applications 125 21. Forms orthogonal to holo~orphic forms 125 22. Generalization of Theorem 8.1 130 23. Weighted formula for solving the {}-problem in strictly convex
domains and zeros of functions of the Nevanlinna-Dzhrbashyan class 132
CHAPTER Vll. Representation and multiplication of distributions in higher dimensions 137 24. Harmonic representation of distributions 137 25. The product of distributions and its properties 141 26. Examples of products of distributions 144
Supplement to the Brief Historical Survey 151
Bibliography 153
Index of symbols 165
PREFACE TO THE AMERICAN EDITION In the six years that have elapsed since this book appeared in the USSR,
many new results have been obtained in this field of multidimensional complex analysis. These results are presented in a supplement (Chapter V-VII), written by the authors especially for the American edition. The results of A. M. Kytmanov have niade the greatest impact on the contents of the supplement. He has also written Chapter VII of the supplement at the request of the authors. We take this opportunity to thank him for this work; we also thank him and S. G. Myslivech for help in preparing the manuscript of the supple ment.
vii
PREFACE In this book we consider the problem of characterizing the exterior differen
tial forms which are orthogonal to holomorphic functions (or forms) in a domain D c en with respect to integration over the boundary, and some related questions. We give a detailed account of the derivation of the Bochner Martinelli-Koppelman integral representation of exterior differential forms, which was obtained recently (1967) but has already found many important applications. A complete proof of this representation has not previously been available in our• literature. We study the properties of a-closed forms of type (p, n- 1), 0 or;;; p ~ n- 1, which tum out to be the duals (with respect to the orthogonality mentioned above) to holomorphic functions (or forms) in several complex variables, and resemble holomorphic functions of one complex varia ble in their properties. At the end of the book, we ~ve some applications, in particular to the problem of multiplying distributions, and also a brief histori cal survey and a discussion of open problems.
We hope that this little ·book will be useful to mathematicians and theoreti cal physicists interested in several complex variables.
The greater part of the results expounded below were obtained by us during the years .1970-73. They were reported in seminars at Krasnoyarsk, Moscow State University, Urals State University (Sverdlovsk), and the Institute of Mathematics in the Siberian Division of the Academy of Sciences of the USSR. We thank the participants of these seminars for useful discussions. Above all we are grateful to G. M. Khenkin and V. P. Palamodov for valuable remarks.
• Editor's note. The authors mean "Russian".
ix
INTRODUCTION A large number of results in the theory of holomorphic functions of one
complex variable depend on the following, already classical, assertion:
Let D be a bounded domain in the plane with smooth boundary aD, and cp(z) a continuous function on aD. Then
1 f(z)cp(z) dz = 0 ilD ,
{0.1)
for all functions f( z) continuous in the closed domain i5 and holomorphic in D if and only if cp(z) can be extended holomorphically into D (s~. for example, Mushelishvili [1], §29). Briefly, holomorphic functions, and only these, are orthogonal to holomorphic functions.
By attaching the dz in (0.1) to f(z) or to cp(z), the same result can be reformulated in the following way, which is more convenient for our exposi tion: a continuous (on aD) exterior differential form(') a of type (p,O) is orthogonal rmder integration on aD to all forms of type (1 - p, 0) which are continuous in i5 and holomorphic in D, if and only if there exists a form y of type ( p, 0) holomorphic in D and continuous in i5, such that a = y lao· Here p = 0 or 1.
The analogous problem is of great interest in the theory of functions of several complex variables:
What forms a, defined on the boundary of a domain D in C" with smooth boundary, are orthogonal to holomorphic forms 11 (in particular to holomorphic functions) in the sense that
1 a/\t~=O i!D
for all forms holomorphic on i5 (or holomorphic in D and continuous on D)?
( 1) In the sequel, we shall simply write "form" instead of "exterior differential form".
1
{0.2)
2 INTRODUCTION
Kohn and Rossi [1] have considered the more general problem of describing forms which are orthogonal to all a-closed forms of type (n- p, q). They have shown that, if q > 0, and if at each point of aD the Levi form has q + I positive eigenvalues or n - q negative ones, then a CIX) form a of type (p, n- q- I) on aD is orthogonal (in the sense of (0.2)) to all a-closed CIX) forms of type (n - p, q) on D if and only if there exists a CIX) a-closed form y
of type (p, n - q- I) on i5 such that y l1w = a. However this solution does not include the case of interest to us, since
holomorphic forms are a-closed forms of type ( p, 0). It will be proved (Chapter II) that a-closed forms of type (p, n- 1),
0..;;; p..;;; n, are the duals; with respect- to integration over the boundary of the domain (in the sense of (0.2)), of holomorphic forms of type (n- p,O); in particular, for p = n, they are the duals of holomorphic functions.
The proof makes essential use of the Martinelli-Bochner-Koppelman integral representation of exterior differential forms, as also the "barrier function" of Khenkin for strictly pseudoconvex domains.
To state the result in a concrete way: forms which are orthogonal to all holomorphic forms on the closure of a strictly pseudoconvex domain must be extendable to the interior as a-closed forms, or (what amounts to the same in the case of strictly pseudoconvex domains) as a-exact forms of type (p, n- 1).
On the other hand, it will be established in Chapter III that the a-closed forms of type ( p, n - 1), 0 ..;;; p ..;;; n, are the analogs for en of the holomorphic functions of one complex variable, i.e. they retain several of the important properties of these functions which e.g. the holomorphic functions of n variables, n > 1, do not possess.
Chapter I is devote4 to tools that will be used and results related to the main problem. In particular, we present an elementary derivation of the Martinelli Bochner-Koppelman integral representation of forms, generalizing the Martinelli-Bochner integral representation for holomorphic functions on the one hand, and the Cauchy-Green formula for smooth functions in the case n = 1 on the other. Also we give a characterization of the trace of a holomor phic function on the boundary of a domain in en.
In Chapter IV, some applications are indicated: we generalize the classical theorems of Hartogs and the Rieszes, describe the general form of integral representations of holomorphic functions, and construct the Martinelli-Bochner representation of a distribution lying on 6j)'(R2n-t), by means of which we introduce a new definition for the product of distributions.
In the text itself, we shall refer only to such original works as are necessary for understanding the book. All references are listed at the end of the book. After Chapters IV and VII the reader will also fmd a brief historical survey; in
INTRODUCTION 3
the former some unresolved questions (as of 1975) are indicated (a few of which are subsequently solved in Chapters V-VII, written in 1981 for the American edition).
We shall employ the following notation. C" is the space of n complex variables, whose points are denoted by z, r. z0, ro. etc. If z = (z., ... ,z,) then z = (Z1, ••• ,z,). For z, r E C" we define (z, 0= z1r 1 + · · · +z,r,, and
lzi=J(z,z), ( ) f- z ( f. - z. f, - z, ) 1 r • z = 1 r - z 12 = 1 r - z 12 • • • • • 1 r - z 12 •
B(z0 , r) = {z E C": lz- z0 1< r}.
For vectors I and J with nonnegative integral components,
aiii-JII DI·J= . . . . .
azj• ... az~·az(• ... az~·
~J. denotes summation over J = (j1, ••• ,jp) with 1 <j1 < · · · <jP < n. Also,
dz = dz1 1\ · · · 1\dz,, dzJ =,dzj1 1\ · · · 1\dzj,•
dz[J] = dz;1 1\ · · · 1\dz;.-,• i 1 < · · · < i,_P'
k= l, ... ,p, l=l, ... ,n-p.
The symbol [ ] will often be employed in the above sense; e.g. z[k] =
(z., ... ,zk-l• zk+t•· .. ,z,). Let M and N be sets in C". Then aM is the boundary of M, and we define
p(z,M)= inflf-zl, Mr={zEM:p(z,aM}>r}, !EM
diam M = sup I r - z I . r,zEM
CM denotes the complement of Min C". M is the closure of M, int M is the interior of M, and M cs: N means that M
is compact and contained in int N. Let F be a closed set and n an open set in C". For m = 0, I, ... , oo,
0.;;;;;. >..;;;;;. I, 0.;;;;;. r.;;;;;. 2n, and 0.;;;;;. p, q.;;; n, crm·"(F) denqtes the class of forms of dimension r whose coefficients extend continuously to em functions in a neighborhood of F such that all derivatives of order m (if m < oo) satisfy a Holder condition with exponent >..
C(~:~>(F) denotes the subset of CP~~(F) consisting of fo~ of type (p, q). For>.= 0 we shall omit the index, and regard C<~.q)(F) = C(~:~>(F) as a
space with the topology of uniform convergence of all coefficients together with a.tl derivatives of order < m on F.
4 INTRODUCTION
c<;:~>(D) denotes the projective limit of the classes C(;:~>(F), F ~ D, with the projective limit topology.
If
then
a= ~' audz1 1\ dz1 E c<;:~>(D) /,J
- - ~' ~ aalJ -aa- ..(J 41 ydzk 1\ dzrl\ dzJ, I.J k=l zk
Z<';'.~>(D) = {a E c<;:~>(D): aa = 0} (for m = 0, differentiation is to be understood as that in the space of distributions).
Here again, if m = 0 we shall omit the index, and write e.g. Z<~.q)(D) = Z(p,q)(D). Similarly we shall drop the indicies p and q if p = q = 0.
For all a E C(p,q)(D) we set supp a= {z ED: a(z) =1= O},e) and
AP(D) = Z(p,o)(D) = Z(;,o>(D),
ACP(D) = Z<p.o>(O) = AP(D) n C(p,o>(O).
Ap(K) denotes the inductive limit of the spaces Ap(D), D::) K, where K is a compact set in en.
If a E c<;,q)(D) and fP E C00(en) with supp fP ~ D, then we shall suppose that qJa is extended by zero to CD, so that qJa E c<;.q)(en).
The index z (or n under a domain of integration signifies that z (respectively n is the variable of integration.
We shall say that Dis a domain with boundary of class cm,ll. if D = {z E en: p(z) < 0}, where p E C'"·ll.(en) and grad p =F 0 on 3D.
If m = 1 and A = 0, D is said to have smooth boundary.
erThe bar denotes closure.
AND ITS IMMEDIATE CONSEQUENCES
§1. The Martinelli-Bochner-Koppelman formula
1 o. The definition of the determinant of a numerical matrix uses only the operations of addition and multiplication; hence we can form the determinant of a matrix whose elements lie in an arbitrary ring.
A determinant of order n is the algebraic sum
all a,2 a,n
ani an2 ann
where e1 is the parity of the permutation I= (i1, • •• ,in). Thus the determinant is expanded according to the usual rule, the place where a factor occurs in the summand being determined by the column to which the factor belongs.
In connection with the expansion of a determinant, it is convenient to introduce the following notation.
Let 0 I' .•. '0 m be n-dimensional vectors. Then
v ...... ,. .. (o', ... ,om)
is the determinant of order n = v1 + · · · +vm whose first v1 columns are 01,
the next v2 columns are 02, and so on. We state some properties(') of determinants which will be needed subse
quently.
( 1) These properties are trivial from the point of view of algebra; however for some of them we have not been able to fmd appropriate references, and have taken the liberty of listing them for the convenience of the reader.
5
6 I. INTEGRAL REPRESENTATION OF FORMS
PROPERTY 1. If each element of the kth row (or column) of a determinant has been expressed as the sum of two terms, then the determinant is equal to the sum of two determinants having the same rows (columns) as the given determinant except the kth, which consists of the first and second summands respectively.
PROPERTY 2. If two rows of a determinant are interchanged, then the determi nant changes sign.
PROPERTY 3. If all the elements of a certain column commute with all the elements of an adjacent column, then the interchange of these two columns changes the sign of the determinant; if they anticommute,e) then the determinant is unaffected by the interchange.
PROPERTY 4. If all the elements of a column (or row) are multiplied by an element which commutes with all the elements of the determinant, then the determinant also gets multiplied by this element.
PROPERTY 5. If all the elements of the first column are multiplied on the left by a certain element, then the determinant also gets multiplied on the left by the same element.
For the remaining properties which we state, we assume that the entries of the determinant lie in the ring of double forms (see de Rham [1], Chapter II, §7). Observe that Property 4 yields
PROPERTY 4'. If all the elements of a column (or row) are multiplied by the same function, then the determinant gets multiplied by this function.
PROPERTY 6. If the rows of the determinant are linearly dependent,(l) then the determinant vanishes.
The proofs are the same as for the usual determinants. For forms, and hence also for double forms, we have the following assertion: If the a;, i = I, ... ,m, are forms of class C1, of degree q; in z, then
e) By anticommutation we mean the relation ab = -ba. ( 3) The rows are said to be linearly dependent if some linear combination of them with functions
as coefficients vanishes identically, and the coefficients do not have a common zero.
§I. THE MARTINELLI-BOCHNER-KOPPELMAN FORMULA 7
From this we get
PROPERJ:Y 7. If the elements of the ith columns 8; of a determinant are C 1
double forms of degree q; in z, i = 1, ... ,n, then
n = ~ (-l)q•+···+qi-•D (8' o;-ra 8; Oi+l on) ~ 1 •... ,1 , ••• , z ' , ••• , • i=l
• - ;_- I - ; Here and m the sequel, azo - (az81, ••• ,azOn). 2°. Let w(r, z) = (w1, ... ,wn) be a vector-valued function defined on U c
Cf X c;, with the property that (w, r- z)= 1. If wj E C 1(U) for allj, we let
(-l)q+p(n-q-1)( n- 1) ( ") q (- -) ~.q w, ), z = ( .)n ( ) D1,q,n-q-l w, a.w, arw
· 2'1Tr p! n- p !
1\Dp,n-p(az, ar).
Here the subscripts z and r of a indicate the group of variables with respect to which a iS acting, and az and ar denote the Vector forms ( dz 1, ... , dz n) and (dr1, ••• ,drn> respectively.
We also set ~.-I = ~.n = 0. The kernels of the integral representation for exterior differential forms will
be the forms ~.q(w, r. z), for a special choice of the vector w, namely w = t(r, z) = (r- z)/1 r- z 12• We therefore introduce the notation
~.q(r. z) = ~.q(t(r, z), r. z).
Observe that JJ'0,0(w, r, z) is the Cauchy-Fantappie kernel; for this form we shall also write
"'( w, r. z) = Wo.o( w, r. z ).
We can now state the Martinelli-Bochner formula for exterior differential forms:
Let D be a bounded domain with piecewise smooth boundary, and let y E I -
C,.p,q)(D). Then
1 ( )- 2( - )--2 -{y(z), Ip,qD,y Ip,qD,ay 3Ip,q- 1(D,y)- O, zED,
z ~ 15, (1.1)
where
I},q- 1(D, Y )(z) = JD y(r) 1\ ~.q- 1 (t z ). r
The proof of (1.1) will be given in 5°. This formula is valid in greater generality. For example, we could assume that Dis a finite union of domains with piecewise smooth boundary, whose closures are pairwise disjoint.
Also, for q = n, (1.1) holds for any open set D and any form y E C<~.n>(D) whose coefficients are absolutely integrable in D. Indeed, let B be a ball with B <&D. Since (1.1) holds forB, we have for z E B
For z E B, we may differentiate the second integral under the integral sign. Now az~.n-l = 0 by Lemma 1.2; hence (1.2) reduces to (1.1) for q = n, under the assumptions made on D and y.
3°. Observe that
D1,q,n-q- 1(t, azt, a,t) I - I (- - - )
= I r- z rz az I~- z 12 1\ D2,q-l,n-q-l ~- i, azt, a,t
- I r ~ z 14 Dl,l,q-l,n-q-l(f- i, azi, azt, a,t ) . .
The first determinant on the right has two identical columns; hence it vanishes, by Property 3 of ~terminants.
Repeating this procedure q - I times more, we get
( - - ) (-I)q ( D tat at= D ------ l,q,n-q-1 'z' r ~~-zl2q+2 l,q,n-q-1 r z,azz,a,t).
§I. THE MARTINELLI-BOCHNER-KOPPELMAN FORMULA
Proceeding analogously with the last n - q - 1 columns, we get
D1,q,n-q- 1( t, az1, a,t)
(-l)q (- -----) = lr-zi2"Dl,q,n-q-l r-z,azz,a,r
9
= (- 1)q~~ (n -12~- I)!~' ± a(l, k)(~- zk)di1df [I, k], (1.3) - Z I k=l
kf/.1
where I= (i1, ••• ,iq) is a multi-index, the prime over the summation sign indicates that the summation is over increasing multi-indices, k f£ I means that k is different from i 1, ••• ,iq, and the constant a(/, k) is determined by the equality
Since
Dp,n-p(az, ar) = p!(n- p )! ~' a(J)dz1 · dr[J], J
we deduce from (1.3) that
( l)P(n-q-1)( l)l n
kf/.1
dz1 I\ di[J] = a(J)dz.
For the proof of (1.1), we need one more form of expression for Uo,q· Let
vq = (- 1 ~:(n.;: 1)! ~' ± a{I, k)(-1)k-ldf [I, k] I\ dr[k]dzl; '1Tl I k=l
kf/.1
then
(-1t+l 1 u = a AV. O,q 1 - n r I r - z 12n-2 q
(1.5)
4°. Throughout this subsection, the letter w (possibly with superscripts) denotes a vector-valued function w(r, z) = ( w.<f, z ), ... , w"(r, z )), where (r, z) E U C (Cf' X Cz")\{(r, z): r = z}, and (w, r- z)= 1.
LEMMA 1.1. The determinant
( 1-2 -,-,,-) D 1 •... ,t w , a.w , ... ,a.w, a,w + , ... ,a,wn
remains unaltered if the vector w1 is replaced by any other w defined in the same domain.
PRooF. Subtract from the given determinant the new determinant
( -2 -,-,,-) D1, ••• ,t w, a.w , ... ,a.w, a,w + , a,wn .
By Property 1,
( I - 2 - I - l+l - ) D 1, ..• ,t w - w, a.w , ... ,a.w, a,w , ... ,a,wn . Since (w1 - w, r- z)= (a.w;, r- z)= (a,w;, r- z)= 0, we see by Prop erty 6 that this last determinant vanishes.
LEMMA 1.2. If w is of class C2 and 0 EO; q EO; n, then
- ) - ( p+q-a,Uj,_q(w, r. z - -1) a.Uj,,q-l(w, r. z).
In particular, a,Uj,,o = a.Uj,,n-t = 0.
PRooF. For I EO; q EO; n, we have by Property 7
a.Dt,q-t,n-q( w, a.w, a,w) = Dq,n-q(a.w, a,w)
(1.6)
+ (-1)q- 1(n- q)D1,q-t.t,n-q-t( w, a.w, a.a,w, a,w ). (1.7)
For 0 EO; q EO; n- 1, the same property yields
a,D,,q,n-q-t( w, a.w, a,w) = (-1)qDq,n-q(a.w, a,w) +qDt,q-t,t,n-q- 1( w, a.w, a,a.w, a,w ). (1.8)
By Property 6, the first determinant in each of (1.7) and (1.8) vanishes. For q = n in (1.7), and for q = 0 in (1.8), the second term vanishes; hence from the definition of Uj,,n-t and Uj,,n we see that (1.6) holds for q = 0 and n.
For 1 EO; q EO; n - 1, the second determinants in (1.7) and (1.8) are equal to each other, since a.a,w = a,a.w. Here a.a,w is regarded as a double form. Therefore
a,[ (n- q )D1,q,n-q-t( w, a.w, a,w )] = ~.[(-l)q-tqD1 ,q-t,n-q( w, a.w, a,w )].
Multiplying this equality by
(n- I)! (-I)P(n-q-1) _ _,___,___,_---'----D _ ( az ar) (2witq! (n- q)!p!(n- p )! p,n P ' '
we obtain (1.6).
LEMMA 1.3. Ifw 1 and w2 are vector-valued functions of class C2 in~. then
( 2 )' ( 1 )-- 1 »-;,,0 w,~,z -»-;,.0 w,~,z -a,p.0 /\p!(n-p)!
XDp,n-/az, an, where
P.o- ( .)n ""' Dl,l,j,n-j-2 w 'w 'a,w' rW . 2'1Tl j=O
In particular,
The proof is a verification, using Properties 3 and 7 and Lemma 1.1. Similar considerations are used in the proof of the next lemma.
LEMMA 1.4. If U C Cf X Czn is such that there exists a vector-valued function u(~, z) E C2( U) holomorphic in z for fixed~ and satisfying ( u, ~ - z) = I, then for all wE C2(U)
- (- I - 2) I ( ) ( 1 9) ~,q(w,~,z)- azP.q-l +asP.q !\ p!(n-p)!Dp,n-p az,a~, ·
where p.~ and p.~ are forms which are linear combinations of forms of the type
D 1,1,q,r,n-q-r-2( w, u, azw, a,u, a,w ). (1.10)
PRooF. We first show that
( - - ) -- q-1 D1,q,n-q- 1 w, 3zw, a,u - azan-q- 1 (1.11)
and that for 0 .;;; r .;;; n - q - 2 there exist constants c, such that
D 1,q,r,n-q-r-l( w, azw, a,u, 3rw)- c,D1,q,r+l,n-q-r-2( w, azw, 3ru, 3rw) = aza;-l + a,{J,q, {l.I2)
where a; and {J,q are forms differing from (I. I 0) by a constant factor. Since u, and hence a,u, is holomorphic in z, by Property 7 we have
azDI,l,q-l,r,n-q-r-1( u, w, azw, a,u, a,w)
= D 1,q,r,n-q-r-l( u, azw, a,u, a,w) + (-I)q- 1(n- q- r- I)Dl,l,q-l,r,l,n-q-r-2 (1.13)
X ( u, w, azw, 3ru, azarw, a,w ).
For r = n - q- 1, the second term on the right side of (1.3) disappears, and the first term is equal to the left side of ( 1.11) by Lemma 1.1 ; hence ( 1.11) holds.
For 0 E;; r E;; n - q - 2, we have
arD1•1.q.r,n-q-r-i u, w, azw, ar"· a,w)
= ( -1)q-l D1,q,r+ l,n-q-r- 2( w, azw, a,u, arw) + ( -1)q Dq,q,r,n-q-r- 1( u, azw, aru, arw) +qD1,1,q-l.l,r,n-q-r-2( u, w, azw, a,azw, aru, arw ).
Hence by (1.13) and Lemma 1.1 we get
azqD1,1,q-l,r,n-q-r-l( u, w, azw, a,u, arw) +a,(-1)q(n- q- r- t)D1,1,q.r.n-q-r-i u, w, azw, a,u, a,w)
= (n- r- 1)D1,q,r,n-q-r-l( w, azw, a,u, a,w)
- (n- q- r- 1)D1,q,r+l,n-q-r-2( w, azw, aru, arw ).
Dividing both sides by n - r- 1, we obtain (1.12). To prove (1.9), we now need only multiply (1.12) by c0 • • • • ·c,_ 1 and sum
over r, and then add (1.11) multiplied by c0 • • • • • c,_q_ 2•
LEMMA 1.5.
PRooF. We have
(-l)q+p(n-q-1){ n; 1) ~.q(~. z) = (2 ·)" I ( - )' .,., p. n p .
XD1,q,n-q-l( t(~, z ), azta, z ), a,t(~, z)) 1\ Dp,,-p{az, ar),
(-l)"-q,-l+(n-p)q{ n; 1.) ll,-p,n-q-l(z, n = ( ")" ( ) 2'1Tl p!n-.p!
XD 1,,-q-l,q(t(z, r), a,t(z, r), azl{z, r)) 1\ D,-p,p(a~. az).
The required result now follows by Property 3, if we take into account the relation t(~, z) = -t(z, n.
§1. THE MARTINELLI-BOCHNER-KOPPELMAN FORMULA 13
LEMMA 1.6.
w(w, r. z) = (n- l~! ~ (-l)k-lwkdw[k] 1\ dr (2wi) k= 1
_ (n- 1)! (-l)k- 1dw[k] 1\ dr - (2wir - rk- zk
where the Second equality holds for rk =fo Zk.
PRooF. To prove the first equality, we need only expand the determinant D1,,_ 1(w, arw) by elements of the first column and observe that Dn<an = n!dr. To prove the second equality, we use Lemma 1.1 to replace the first column of D1,,_ 1(w, a,w) by the column whose kth component is <rk- zkt1 and all other components are zero, and expand the resulting determinant according to elements of this column.
5°. PRooF OF ( 1.1 ). We need the following lemma.
LEMMA 1.7. Iff E C 1 in a neighborhood ofO E C", then
. 1 f(r)df 1\ dr[k] {(-t)"-k-l <2'1Ji)" at (o)· hm = n! ar . e-O lfl=e I r 12n-m k
0; m>O.
m=O, (1.14)
In particular,
lim 1 g(r)rjdf ~ dr[k] = (-I)"-k-1 (2w;r g(O)·Bjk· e-o lfl=e I r I " n.
Here 6jk is the Kronecker symbol ( B1k = 0 for j .P k, and 81j = 1).
PROOF. We have
1 f(r}df 1\ dr[k] = -•-1 J(r)df 1\ drfk] I.CI=e I r 12"-m £2"-m lrJ=e
= (-l)"-k-l.em-2n1 at df 1\ dr lfl<e ark
= ( -1 )n-k-1. em-2n[ at (0)1 df 1\ dr + 1 ( at <n- at (o)) df 1\ dr]. ark I.CI<e lrJ<e ark ark
(1.15)
1 df 1\ dr = (2w;)" e2",
lfl<e n.
so that the first term on the right side of (1.15) gives the right side of (1.14). But the limit of the second term as e .... 0 is zero, since the integrand tends to zero as e .... 0. So the first part of the lemma is proved.
To prove the second part, we need only observe that
a ark (grj)(o) = g(o)Bjk·
We now proceed to the proof of (1.1). In view of the additivity of the operators on the left side of ( 1.1 ), we need only prove the formula for forms of the type
y = /( z )dz1 1\ dzJ.
Here/ E C1(D), I= {i1, ... ,iq) and J = (j1,. .. ,jp). Now
(-l)q+p(n-q-1)( n; 1) _
y(r) 1\ Up.q(r, z) = (2 .)n I ( _ )I f(r)drl 1\ drJ 7TI p. n p .
I\D1,q,n-q-t(t, azt, art) 1\ Dp,n-p(az, or}
_ (-l)q( n; 1} _ _ = f(r)drl 1\ (27Tir D1,q,n-q- 1(t, ozt, ort) 1\ dr 1\ dzJ
= f(r)dfr 1\ Uo,q(r, z) 1\ dzJ,
and analogously
i}y 1\ ~,q(r, z) =a( /(r)dfr) 1\ Uo,q 1\ dzJ,
y 1\ ~.q- 1(t z) = f(r)dfr 1\ Uo,q-l 1\ dzJ.
these equalities show that (1.1) needs only to be proved for forms of type (0, q), or more precisely for forms of the type y = f(z)d"i1•
Finally, the kernels Uo,q are invariant under the transformations ri .... r .. <i>• zi .... z,.(j),j = 1, ... ,n, T being an arbitrary permutation. Indeed, under such a transformation, the same rows of the two determinants occurring in the expression for Uo,q change their places. Thus it suffices to prove (1.1) for the forms of the type
y = !( z )dZ, 1\ ... 1\d"iq.
For z f£ D, (1.1) is an immediate consequence of Stokes' formula and Lemma 1.2.
Letz ED. We set
-II 12-211 g=(l-n) ~-z ,
Since y 1\ Uo,q has type (n, n...,. 1) in ~. and y 1\ azg 1\ Vq-l is of type (n- 1, n) in~. we have
d~( y 1\ o;) =a,( y 1\ U0,q)- ar( y 1\ azg 1\ Vq-l)
= arY 1\ Uo,q + (-l)qy 1\ arUo,q- arY 1\ azg 1\ ~- 1 q -
-(-1) y 1\ a!'azg 1\ vq-l•
As we already observed, azarg = arazg; hence it follows from (1.5) and Lemma 1.2 that
(-l)qy 1\ arazg 1\ vq-l = (-l)qy 1\ az(arg 1\ vq-l) - q -
= y A apo,q-l = (-1) y A a,uo.q·
This together with the preceding relation yields
dr( y 1\ o;) = arY 1\ U0,q- arY 1\ azg 1\ Vq-l· {1.16}
Therefore the second integral in
{1.17)
is absolutely convergent. We shall now show that i(z) is precisely the left-hand side of (1.1). For this
we consider the form y 1\ gVq-J· This is of type (n- 1, n) in ~; hence by Stokes' theorem
1 y 1\ gVq-l -1 y 1\ gVq-l aDr <lr-zl=e}r
where D~ = {~ E D II~ - z I> e}. Lemma 1.7 shows that the second integral on the right in (1.8) tends to zero as e-+ 0, so that (1.5) and (1.18) yield
!, Y 1\ Uo,q-1 = 1 y(r) 1\ 8~-l-!, ay(r) 1\ gVq-l· (1.19) Dr aDr . Dr
In both the integrals on the right side of (1.19), we may differentiate under the integral sign; the first is a proper integral, while the second has a singularity of the form 1 z- r p- 211 arter differentiating the integrand and
hence is absolutely convergent. Hence, the left side of (1.19) is likewise differentiable, and we get
1 y(r) 1\ azg 1\ Vq-l- f ()y(r) 1\ azg 1\ Vq_,-a j y 1\ Uo,q-l = 0. aD! Dr Dr
(1.20}
Adding (1.17) and (1.20), and using the definition of Uo,q and (1.16), we obtain
We shall show that i(z) = y(z). From (1.17) and Stokes' theorem applied to D, we get
i(z)=IimJ, y(r}/\8;. e-+0 1r-z1=e
Let us compute this limit. To do this, we go over from 8; to the form
From the invariance of the first differential, it follows that drg IK = 0, where K = {r E en: I r - z I= e), or a,g IK = a,g IK; hence (1.5) and the definition of 8; show that 8; IK = 8q IK·
We set
y(!} 1\ 8 = (-l}n (n- 1}! /(r} q (2tri)n
X {(-l)qdf1 1\ • · • 1\dfq 1\ I gkdfk 1\ ~~ ~ a(I, j)(-1)1- 1df [I, j) k=l I jf/.1
n
Adr[j]dir- df, A .. · Adfq A ~ (-gk)dik • j=l
1\~1 ~ a(J, k)(-l)k-t X df [J, k) 1\ dr[k)diJ} J kfl.J
§1. THE MARTINELLI-BOCHNER-KOPPELMAN FORMULA 17
~{t a((l, ... , (j], .. .,q, k ), j)( -I )1- 'dl'(j] 1\ az, 1\ ... (j]
.. · az, 1\ az, + a(( I, ... ,q ), k )(-I),_, dl'[ k] 1\ az, 1\ .. • 1\di,]
q
+d~l\···l\dfql\df[I, ... ,q1 1\ ~ o({l, ... ,[k1, ... ,q),k) k=l
Observe also that
(-I)qo((I, ... ,q); k)df1 1\ · · · 1\dfq 1\ dfk 1\ df [l, ... ,q, k1
= o({l, ... ,q), k)dfk 1\ (df1 1\ df2 /\ • • ·1\dfq) 1\ df [l, ... ,q, k1
= df, o((l, ... , [k], ... ,q), k)df1 1\ · · · 1\dfq
1\df [I, ... ,q1dZk l\dZ1 1\ ·· · [k1 · · ·1\d"iq
= o((l, ... , [k1, ... ,q), k)dfk
Adf1 1\ · · · [k 1 · · ·1\dfq 1\ df [ 1, ... ,q ]dz• 1\ ... 1\dZq = dfdz• 1\ · · ·1\dzq.
Using these equalities, we get
y 1\ O = (-It{n- I)! /(r) ~ {-l)k-Ig df q (2 ")" ~ k 'tTl k=l
1\dr(k 1 1\ dZI 1\ · · · 1\dZq +X·
Here the form x is a sum of terms of the type
Thus finally we get by Lemma I. 7
i(z ). =lim 1 y(r) 1\ o: =lim 1 y(r) 1\ oq e-O 1r-zj=e e-O lr-zl=e
= (n- I)! ± lim (-l)n+k-11 f(r)(tk- zk) df (2'11'ir k= I e-O lr-zl=e It- z l2n
Adr( k] A dz1 A · · · 1\tliq + lim 1 x e-o lr-zj=e
(n- I)! ~ (2'11'irf( )J: 1\ 1\d- _ ( ) = ~ -- z uz1 · · · z - y z . (2'11'ir k=l n! q
§2. 1lleorems on the saltus of forms
1°. In the theory of functions of one complex variable, the following results are well known (see, for example, Muskelishvili [1], §22).
PROPOSITION 2.1. Let D be a domain in C1 with smooth boundary, and let 'P E C0·'-(aD). Then the functions
'P±(z) =-1 ·1 'P(r}dt 2'11'l i!D t- Z
( 'P + is defined in D, and qJ- in CD) can be extended continuously to the closure of the open set on which they are defined, and
'P+ <n- 'P- <n = 'P(r), reaD. (2.1)
To formulate the second proposition, we need some notation. Let D be a domain with smooth boundary, and let 'P E qaD) and z0 E aD.
On a line v passing through z0 an<J not tangential to aD, choose points z E D and z' fl. i5 at equal distances from z0 , and consider the limit
lim ['P+(z)-'P-(z')] {2.2) z, z',....z0 ·
( 'P± are defined as in Proposition 2.1).
PROPOSITION 2.2. The limit (2.2) exists and equals qJ(z0). The limit is approached uniformly if the nonobtuse angle between the line v and the tangent to aD at Zo is not less than a fixed Po > 0.
It must be noted that Proposition 2.1 solves the so-called additive Riemann problem:
Let D be a domain with smooth boundary, and let 'P E C0·>-(aD). Find functions 'P+ E AC(D) and (f}-E AC(CD) such that 'P+ -qJ-= 'PonD.
§2. THEOREMS ON THE SAL TUS OF FORMS 19
We shall need analogues of the Riemann problem and Propositions 2.1 and 2.2 for forms of several complex variables. .
2°. Before proceeding to consider the analogues mentioned above, we state the following result without proof.
THEOREM 2.3. Suppose aD E em+ I,A and IE em·"( D), where m ;> 0 and 0 < A EO; 1. Then the functions defined by the integral
1 t<ndf [k1 1\ dt aol lt-zl2n-2
(2.3a)
inside and outside D can be extended to the closure of the respective open sets as functions of class em+ I,A', and the junctions defined by the integral
(2.3b)
as functions of class em+l,>.', 0 <A' <A.
The proof of this theorem can be carried o~t along the lines of Gyunter's proof ([1], Chapter II, §19) for the case of real dimension three. But the computations are much more complicated in our case.
Since the coefficients of the kernels ~.q are derivatives of the function e 1 r- z 12- 2n, and we may differentiate under the integral sign in (2.3a) and (2.3b ), we obtain
COROLLARY 2.4. If aD E em+I,A andy E G~:~)(D), then the forms defined by the int~al I(~,q)(D, y) inside and outside D extend to the closure of the respective open sets as forms of class<;~:~;. and the forms defined by I(~,q-l)( D, 'Y) extend as forms of class e(~~~~;. with 0 < A' < A.
3°. We observed that (1.1) is the analogue of the Cauchy integral formula for forms of several complex variables. Therefore it is natural to consider an analogue of the integral of Cauchy type an integral of the form I;,q(D, y)(z). Here Dis a domain with piecewise smooth boundary, and y E <;p,q)(aD). Let y+ denote the form defmed in D by the above integral andy- the one defined in eD.
THEOREM 2.5. If aD E e2 and 'Y E G~.q)(aD}, 0 EO; q.;;;; n - 1, then
'Y = 'Y+Iao- 'Y-Iao
PRooF. It suffices to prove that
(2.4)
for any q> E C~-q-t(aD). We extend q> to C" as an (n- q- 1)-form of class C 1• Then q> can be represented in the form
n-q-l
qJ = ~ fJJ;, i=O
The forms y ± and r are of type ( n, q ); therefore r 1\ fJJ; = y ± 1\ fJJ; = 0 for i = 1, ... ,n- q- 1. Hence we need prove (2.4) only for q> E C(~ ... -q-t)(C").
Let p be a function defining D; thus, D = {z E C": p(z) < 0}, p E C 2(C"), grad p -:1= 0 on aD. Set D1 = {z E C": p(z) + /-1 < 0} and D1 = D_1• Then for sufficiently large I we have
and also, for any X E C2,-t(C"),
1 X= lim 1 X= lim 1 X· aD t- aJ aD, t- aJ aD'
Thus
= lim [1 q>(z) 1\1 y(r) 1\ U,.q(~. z) t-aJ caD,), aDc
-1 , q>(z) 1\1 r(r) 1\ U,.q(t z >] caD), aDc
= lim 1 r(r) 1\[1 q>(z) 1\ (-u, q(~. z)) t- aJ aDc caD'>, •
-1 q>(z) 1\ (-u,,q(t z))l. caD,>, J
The reversal of the order of integration is permissible since the integrands are jointly continuous. By Lemma 1.5, U,,q(~, z) = -Uo,n-q- 1(z, n, Hence by
§2. TIIEOREMS ON THE SAL TUS OF FORMS 21
( 1.1) and Stokes' theorem we get
1 (y+-y-)Acp=lim1 y(r)/\1 cp(z)AOo ... -q-l(z,r) aD , .... <¥) caD>r [a(D1\D1>J,
=lim 1 y(r) 1\ {cp(r) + 1 acp(z) 1\ U0 ,_q_ 1 (z.~) , .... <¥) caD>r (D1\D1>, ·
+a 1, cp(z) 1\ Uo.n-q-2(z, r>} (D \D1),
=1 yAcp+ lim 1 {y(r)/\1 acp(z)I\U0,,_q-l(z,r) aD f .... oo aDr (D1\D1),
+ {-I)"+qay(~) 1\1 cp(z) 1\ U0 n-q-2(z, r)}. (D1\D1), '
From the form of the kernels Uo,r• we see that the last limit above is zero as a consequence of
wheref(r, z) E C(aD X D1\D1).
We show that the integral
1 1 dsdv aDX(D1\D,) I r- z 12"- 1
(2.6)
exists. Here dv is the 2n-dimensional volume element, and ds the area element on aD. Since the integrand is positive, (2.6) exists if and only if the following limit exists:
lim 1 1 dv ds. (2.7) ..... o (aDX(D1\D,)l\{lf-zi<e} I r- z 12"-1
Now (2.7) is a proper integral; hence we may go over to the iterated integal, and the existence of the limit (2.7) is equivalent to that of
lim 1 g.(r) ds, where g.{r) = 1 . 1 _ dv. (2.8) e->0 aD . (D1\D,)\{If-zl<e} I r - z 12" I
But lim .... 0 g(r> exists, and
Oo;;;g(r).r;;;j dv . os;;;f dv <oo, • D'\D, I r- z 12n-l vIz 12n-l
where V = Ureaol(D1\D1)- rJ <&en; hence the limit (2.8) exists by the Lebesgue dominated convergence theorem, and so (2.6) exists too.
Since the integrand in (2.~) is majorized by that in (2.6), we may reverse the order of integration in (2.5), and the function
1 J(r. z)(fk- ik) df (j) 1\ dr aDr I r- z 12n
is integrable; hence the limit in (2.5) equals
lim i di 1\ dz1 J(r. z )(rk- zk) df (j) 1\ dr = 0, t-<XJ CD1\D1>, aor 1r-z12n
since the measure of D1\D1 tends to zero as 1-+ oo. 4°. In this subsection, we shall obtain an analogue of Proposition 2.2 for
forms of type (p,O). Let D be a domain with smooth boundary, z0 E aD, and y E C{p,o)(aD). In the interior of the right circular cone ~o with vertex z0, the normal to aD at z0 as axis, and angle between the axis and generator P < 'IT/2, we choose two points z ED and z' f$. i5 such that a I z- z0 1.;;;1 z'- z0 I.;;; b 1 z- z0 1, where a and b are constants with 0 <a.;;; b < oo. Consider the limit
lim [y+(z)-y-(z')]. z, z'-.z0
(2.9)
THEOREM 2.6. The limit (2.9) exists and is equal to y(z0). The convergence is uniform in z 0 E aD if p, a and b are fixed.
Note that for n = 1 this is a strengthening of Proposition 2.2. PROOF. Exactly as in the proof of (1.1) we can show that it is enough to
prove the theorem for (0, 0)-forms, i.e. functions. Thus let y E C(aD). Then
y+ (z)- y-(z') = IJ,0(D, y(r)- y(z0 ))(z)- IJ,0(D, y(r)- y(z0 ))(z')
+y(z0 )[1J,0(D, l){z)- IJ,0(D, l){z')].
By (1.1), !J.0(D, l)(z) = 1 and IJ.0(D, l)(z') = 0; hence we need only prove that
lim 1 (y(z)- y(z0 ))(Uo,0(f, z)- U0,0(f, z')) = 0. (2.10) z,z'-z0 iiD
By a unitary transformation and a translation, we make take z0 to 0, and the tangent plane to aD at z0 to the plane a= {~ E en: lm ~n = 0}. Then the lefi side of (2.10) is unchanged, and aD will be defmed near 0 by the system of equations 'r = 'w and fn = un + ip(w). Here 'w = (w1, ... , wn_ 1), w = ('w, un) E a, and p E C1(U), where U is a neighborhood of 0 in the plane a
§2. THEOREMS ON THE SAL TUS OF FORMS 23
and p(w) = o(l w 1). Let i and i' denote the projections of z and z' on the Im ~"-axis. Then
lz - il < tan Plil, lz' - i'l < tan Pli'l,
lzl <I il/cosp, lz'l <I i' 1/cosp,
alilfcosP <li'l< bcosPiil.
We fix an e > 0, and choose a (2n- 1)-ball Bin a such that 1)B c U; 2) I y(r(w))- y(O) I< e, wE U;
(2.11)
3) I w - i I..- c I r< w) - z I and I w - i' I..- c I r< w) - z I ' for all w E u and a suitable constant C. Condition 2) is satisfied because y is continuous, and condition 3) because of the relations
lr( w) - wl = IP( w )I= P(lwl), lwl<lw- il
and the similar relations with z' in place of z. Observe that C and the radius of B can be chosen to be independent of z0 E aD.
Let B' ={rEaD: r = r(w), wEB}. Let us estimate the integral over B'. Using (1.4), we get
, _ ( n - 1) ! n k- 1 U0,0(r, z)- U0,0(r, z)- (2 ·)" ~ (-1)
'ITl k=l
Now we use condition 3) and the inequality I r(w) I< Cll w I< Cll w- i I to obtain
fk _ I r- z 1-1 r- z'l 2"-• Irk I I r - z' 12" - I r - z II r - z' I j~o -1 ,-_-z 1-j .....!..1 ,~_:..!....,_z'-12-n--1--j
2n-l ..- clc2n+2 ~
lzl +lz'l (2.12)
We may assume that a1 = a(cosPr1 < 1; then (2.11) implies I w- il:> I w- a1i I and I w- i'l:>l w- a1i I· Thus from (2.12) we get
fk fk <-d Iii I r- z 12" I r- z'l2" I w- ali 12",
(2.13)
where d depends only on a, b, C, C1 and p.
Similarly,
l __ z,_k__ z;. l.s;;; l.zkl + lz;.l .s;;; d) Iii ~~-zl2" ~~-z'l2" ~~-zl2" ~~-z'l2" lw-alil2"' (2.14)
where d 1 depends only on a, b, C, C1 and fJ. Finally,
(2.15)
Here da is the surface element on aD, and ds that on a; d2 is a quantity not depending on z0 E aD. Now, from condition 2) and (2.13)-(2.15) we get
IL, (y(r)- y(O))(Uo,o(~. z)- U0 ,0(t z'))l
.s;;;(n-})!dz{d+dJ)eJ. lilds .s;;;d3e1 lilds. 2'11' B., I w- ali 12" R';..n-1 I w- ali 12"
Here w = (w1,. .. , w,_1, u,.) and i = (0, ... ,0, iy,.); hence I il=ly,. I and
lw- a1il2" = (lwJ 12 + · ·· +lw,._l + u~ + a~y;)". Thus the last integral differs from the Poisson integral only by a constant factor and is independent of i.
To finish the proof, it remains only to note that
lim [ (f(r)- f(O))(U0,0(~. z)- U0,0(r, z')) = 0, z,z'-z0 (<ID\B'lr
and that this limit is approached uniformly since the radius of the ball B does not depend on z0 E aD.
COROLLARY 2.7. If aD E C1•"- and y E C(~~)(ilD), then y = y+ lao - y-lao on aD.
This follows immediately from Theorem 2.6 and Corollary 2.4. We note that, if y E C(ilD), then y ± are in general not extendable to i5 as
continuous functions, and it is not possible to obtain an assertion analogous to Corollary 2. 7 for continuous forms. This is shown by the following example.
ExAMPLE 2.8.( 4) Let D be a domain such that aD contains a (2n- I) dimensional ball B lying in the plane {y,. = 0}, with radius R < 1 and center 0. OnBweset
( 4 ) This example has been constructed in analogy with the one given by Gyunter ([ 1], Chapter II, §7).
§2. THEOREMS ON THE SALTUS OF FORMS 25
Then y E C(B). We extend y as a continuous function on aD. The behavior of the functions y:!: near B is determined by the integral
Let us show that y1 is not bounded in any neighborhood of 0. Indeed, let z = (0, ... , 0, iy, ). Then
C=FO.
The integrals of the other terms vanish since df, 1\ dr, = 0 on B. Furthermore,
,
f 2 A 1R r2"-'dr y1(z)=C1 cos(Tx,)ds 2 2 •
S 0 lin r I · I r + Yn I" (2.16)
Here. S is the (2n- 2)-sphere of radius 1, ds its surface element, ( TX,)A the angle between the x,-axis and the radius vector ofT, and C1 =F 0 a constant.
The first integral on the right in (2.16) is positive; hence
1R r 2"- 1dr 1R r 2"- 1dr IY2( z )I;;;;. C2 o lin r II r2 + y,21" ~ C2 B lin r II ,z + y; I" '
As y, - 0, this integral converges to
1R r 2"- 1dr = lnlln 81 B lin r I r 2" In R '
so that, for sufficiently small y,,
ly,(z)I;;;;.Ic2tnl 1:! I· (2.17)
Since the right side of (2.17) tends to infinity as 8- 0, it follows that y1 is unbounded in every neighborhood of 0.
5°. For p < n and q > 0, the assertions analogous to those of Propositions 2.1 and 2.2 and Theorems 2.5 and 2.6 are false, as the following example shows.
EXAMPLE 2.9. Let D be the same domain as in ~ample 2.8, a = din-q+ 1
1\ · · · /\di,_ 1 1\ dz1 1\ · · · 1\dzP and y = di, 1\ a. Then y E q;,q)(aD). For
z EB,
+I -1 - +I -1 Y ao - Y ao - Yt ao - Yt ao•
where
Yt± = fsry(r) !\ ~)f, z).
Since a(n !\ ~.q is of type (n, n- 2) in f, we have
y(r) A ~.q(r. z) Is= o, i.e. y1± = 0; hence for z E B we have y+ lao- y-lao = 0, but y lao =I= 0.
For such forms, the jump theorem can be obtained in the following way: Let D be a domain with a pi~wise smooth boundary, and y E C(~.q)(aD).
Let y denote any form in C(~.q)(D) such that y lao= y, and let
Y±(z±) = l~)D, y)(z±)- ai},q-t(D, y)(z±),
z+ ED,
Y = Y+lao- Y-lao·
Y+(z) = Y(z) + I}.q{D,a-y)(z)
and for z ~ i5 y_(z) = I}.q(D,a-y)(z).
(2.18)
(2.19)
(2.20)
The integral that occurs _!11 the definition of I},q is absolutely convergent for all z E C"; hence I},q(D, ay) E C{p,q)(C"). Hence the forms Y± can be ex tended continuously to aD preserving (2.19) and (2.10). Substracting (2.20) from (2.19), we get (2.18).
6°. Vf e now proceed to the additive problem of Riemann for forms. We formulate it as follows:
Can every y E G~.q>(aD) be represented in the form
(2.21)
with y1 E Zlp.q/D) and y2 E Zlp.q)(CD), and it not, what are the conditions on y of the validity of such a decomposition?
THEOREM 2.11. I. If aD E C 1 andy E G~.q)(aD), then the deco_mposition (2.21) holds only if there exists y E C(~.q)(C") such that y lao= y and a-y = 0 on aD.
§2. 1HEOREMS ON 1HE SAL TUS OF FORMS 27
II. If aD E cm+t,A andy E cm·'-(aD), m;;;. I, then a sufficient condition for (2.21) to hold is the existence of a 11 E C),q(C11 ) such that 'Y1 IaD = y and ay1 1\ dz1 laD= 0 for any J = (jp+l•· .. ,},.). The forms Yt and Y2 can then be chosen inc(;:~;. 0 < }\'<}\,with Y2 = o(l z 11- 211 ) as I z 1-+ 00.
PROOF. I. We first extend y1 and y2 to all of C11 as forms of class C<~.q)(C 11 ). We continue to denote these extensions by y1 and y2, and set y = y1 - y2. Then Y laD= y, and on aD
a-y = ay. - ay2 = o, since ay1 = 0 for z E jj and ay2 = 0 for z f£ D.
II. We shall show that the forms y ± defined in subsection 5° are a-closed. Indeed,
ay± = a1;,q(D, y) = az1aD y(r) 1\ ~.qa, z) =laD y(r) 1\ az~)r, z). c r
Since y 1\ ~.q is of type (n, n- I) in f, it follows by Lemma 1.2 and Stokes' formula that
ay± = (-l)qlaD/(f) 1\ a,~,q+l(f, z)
= (-l)qlaD ay.(r) 1\ Up,q+t(f, z) = 0. f
For sufficiently large z, the_coefficients of ~,q(t. z) and ~.q-t(r, z) are majorized uniformly in r E D by a function of the form c I z 11- 211. An application of Corollary 2.4 and Theorem 2.10 concludes the proof.
For q = n - I, we can get more information on the forms y1 and y2 occurring in (2.21 ). Let us denote by Bp,n-t( D) the set of a-closed forms of the type
n aG a=~ ~ (-1)k-J a J di[k] 1\ dzJ,
J k=l zk
where the J = (j1, ••• Jp) are multi-indices and the G1 are harmonic functions in D.
THEOREM 2.12. If aD E cm+t,A andy E c<;:!-t)(aD), m ;;a. 1, then (2.21)
holds with y1 E zr;,·.~-•>(D) and y2 E Bp,n- 1(CD) n <;;:!:_ 1>(CD), 0 <'A'< 'A, Y2 = O(lzll-2n)as lzl-+ oo.
lfp = n, then y1 E B,.,,._ 1(D).
PRooF. Let y E C<~:~-t>(D) and y lav = y. Then 1},,-iD, y) E C(~~~1;(CD) (see Corollary 2.4), i.e. there _exists {3 E c<;~~1;(C") such that f3 = 1},,-iD, y) on CD. We set y1 = y+·+o/3 and y2 = 1;,,_ 1(D, y) (y"' have been defined in 5°). We have (see (1.4))
u - a. z) = (n- 1)! L ~ (-1)k-l p,n I ( 2 ')"
'TTl J k=l
hence 1;,,_ 1(D, y) E Bp,,-t(CD). By Theorem 2.9,
Ytlav- Y2lav = Y + lav- ( 1;,,-t(D, Y) - a1},,-2(D, Y))
= Y+lav- Y-lav = Y·
The remaining properties of y1 and y2 are proved in the same way as in Theorem 2.11.
For proving the second part of the theorem, we have to use Theorem 2.5. In connection with Theorem 2.12, we note
COROLLARY 2.13. Let oDE cm+l,A andy E Z['J,:~-t)(D), m;;;.. 1; then y = y1 + 'fJ{3, where y1 E B(p,n-t)(D) n Z['J,·.~-t>(D) and {3 E c<;~!.:1;(D), 0 <"A' <"A.
This follows easily from (1.1) and Corollary 2.4.
§3. Characterization of the trace of a holomcnphic fonn
on the boundary of a domain
1 o. Proposition 2.2 allows us to give a necessary and sufficient condition for a continuous function defined on D to be extendable holomorphically into D (see Muskhelishvili [1], §35). Theorem 2.6, being an analogue of Proposition 2.2, can be similarly used for a similar purpose in the case of functions (or more generally forms) of several complex variables.
THEOREM 3.1. Let D be a domain in C" with smooth boundary, and let
y E C(p,o)(oD). Then a form :Y E ACP(D) such that :Y lav = y exists if and only if
( yi\IJ=O (3.1) lav
for any (J E z~-p,n-l)(D).
§3. TRACE OF A FORM ON THE BOUNDARY 29
PROOF. Necessity. Let D1 be the sequence of domains defined in the proof of Theorem 2.5. Then
1 y 1\ 8 = lim 1 y 1\ 8 = lim j d( y 1\ 8) = 0 iiD 1-~ i1D1 1-~ D1
for any 8 E Z~-p,,- 1{D), since
d(y (\ 8) = a(-y (\ 8) = a-y (\ 8 + (-lY'Y (\ a8 = o. Sufficiency. By Lemma 1.2, iir~.o(f, z) = 0 for f =I= z; ~nee it follows from
the definition of y- (§2, 2°) and (3.1) that y-= 0 for z ~D. Then Theorem 2.6 yields
lim y+(z)=y(z0 ), z-z0 Ei1D
zED
the limit being approached uniformly in z0 if z approaches z0 along the norm to aD. Let us show that the form
_ {y, z E ()D, y-- y+, zED,
' is continuous on i5. Indeed, y E C,-p(()D), and so for each e > 0 there exists 81 > 0 such that, for all z0, z' E aD with I z'- z0 I< 81,
Choose 8, 0 < 8 < 81/2, such that if z' E aD and z lies on the inward normal to aD at z' and I z - z' I< 8, then
jy+ (z)- y(z')l < ej2.
Now let z0 E aD, zED, and I z- z0 I< 8. Choose z' E aD such that p(z, ()D) =I z- z' I· Then z lies on the normal to aD at z', and
Then
I'Y (z) - y(z0 )l.s;;;ly(z) - y(z')l + 1-r(z') - y(z0 )l <e.
Thus we have proved the continuity of y at the boundary points. The continuity at interior points is obvious.
{')By the absolute value of a form, we mean the maximum of the absolute values of its coefficients.
It remains to prove that y E A,( D). For a fixed z ED, on account of Lemma 1.2 we have
=(-It faD/(r) A a,~.1 (r, z).
Let u;(n E C"~..P·"_2(C") be such that U;(n = ~,J(r, z) in some neighbor hood of 3D. Then au;(n E z:_p,rr-I(C"), and (3.1) yields
ay(z) = (-l)p 1 y(r) 1\ au;(r) = 0. a Dr
2°. Suppose f E C1(3D) satisfies the tangential Cauchy-Riemann equations, i.e.
df 1\ dz laD = 0. (3.2)
Then, for any x E Cc.":.rr- 2/D), by Stokes' theorem,
1 tax = 1 df A x = o; aD aD
hence it is natural to call y E Cp,0(3D) a weak solution of the tangential Cauchy-Riemann equations if
1 'Y 1\ ax= 0 for all X E C<":-p 11 - 2>(D). aD ·
THEOREM 3.2. Let D be a domain in C", n ;;a. 2, with smooth connected boundary. Then the space tJ/ weak solutitms of the tangential Cauchy-Riemann equations is identical with the space of traces on 3D of forms in A Cp( D).
The proof is exactly the same as that of Theorem 3.1. We need only show that, if 3D is connected and y is a weak solution of the tangential Cauchy-Rie manil'equations, then y-= 0 in eli.
Let B be a ball with D e& B. Since B is a domain of holomorphy, and
a,~,o(r, z) = 0 for rEB and z f£. B, we have ~.o{r, z) = a,x:(n and Xz E Cc.":-p,n- 2>(B), so that for z f£ B
y-(z) = 1 y(r) 1\ ~.o(r. z) = 1 y(r) 1\ a,xz(r) = 0. (3.3) · aDr aDr
3D being smooth and connected, CD is connected; andy- is harmonic since the coefficients o..!_ ~.o<r. z) are. Hence (3.3) and the uniqueness theorem imply that y-==. 0 in CD.
§4. CASES OF SOLVABILITY OF THE a-PROBLEM 31
§4. Some cases of solvability of the a-problem
1 o. As we know, one of the central questions in the theory of functions of several complex variables is the solution of the a-problem, i.e. of the equation
aa = y, (4.1)
where y is a (p, q)-form, q ;;a. 1. Many questions in complex analysis ar either closely related to this problem, or can be reduced to it. For example, D is a domain of holomorphy if and only if the a-problem is solvable in D for any y E Z(t,,q>(D), 1 ,.;;; p < n (H~rmander [1], §4.2). We need the answers to the following questions.
1. Let y E Zfp.q)(en) and supp y = K. Under what conditions on y and K can (4.1) be solved with the same support?
2. Let y E z<kp,q{ii). ~nder what conditions on D can (4.1) be solved with the same smoothness on D?
2°. To answer these questions, we need the following result of Khenkin [1],( 6 )
sharpened by 0vrelid [1]: Let D be a bounded strictly pseudoconvex domain with boundary of class
em. Then there exist a domain of holomorphy V, D <s; V, a vector-valued function P{f, z), and positive constants e, C1 and C2 such that
1. P{f, z) E cm- 1((V\D) X V), and 2. P{f, z) is holomorphic in z E V, for each fixed f E V\ D. 3. fll{f, z) = (P{f, z), f- z)...P 0 for f E V\D, z E D\{n. 4. I fll(f, z) I ;;;;.c.(p(n- p(z)) + C21 r- z 12
for r E V\D, z E i5, I r- z I< e.
In the rest of this section, we shall denote by u the vector-valued function
==(~ ..... ~). We recall that a domain D c en with boundary of class C 2 is said to be
strictly pseudoconvex if D = {z E en: p(z) < 0}, where p E C 2(en), grad p ...P 0 on aD, and the Levi form
is positive for all z E aD and all w =F 0 such that
ap ap -(z)·w + .. · +-(z)·w = 0. az I az n I n
( 6 ) We state the result only in the form we need.
3°. Let us consider integrals of the form
f. y(z) 1\ D1,t,q,r,n-q-r-2(t, u, a.t, aru, a,t) 1\ Dp,n-p{az, an, D:
f. y(f) 1\ D1,,,q,r,n-q-r-2(t, u, a.t, a,u, a,t) 1\ Dp,n-p(az, an, Dr
where t = <f- i)/1 r - z 12 as usual, and 'Y is a form of the appropriate dimension.
Our interest is in the following question: is it possible, by imposing suitable conditions on y, to extend the forms defined by these integrals to forms on V, differentiable as many times as we need?
Let us first explicitly write down the singularity of the integrals:
LEMMA 4.1.
( - - - ) ~ (r;. - zk)llt·'(f, z) Dt,t,q,r,n-q-r-2 u,t,a.t,a,u,a,t = k~J l(r-z)l2(n-r-t)~r+t'
where the ~£%'' are forms of type (0, q) in z and (0, n- q- 2) in r. with coefficients in cm- 2((V\D) X V).
The proof proceeds exactly like that of (1.4). We can now answer the question asked above.
LEMMA 4.2. Let m ~ 3 and 'Y E cp~; l,A( V). I. Ify = 0 on V\D, then
f. y( z) 1\ D 1,t,n-q,,- 2 ( t, u, a.t, a,u, art) D:
2. lfy = 0 on D, then
1 y(r) A D,,t,q-r,r,n-q-i t, u, azt, a,u, a,t) CV\D>r
1\Dp,n-p(az, ar) E ~~-:;,2_2>(D).
PRooF. We shall prove the first part; the proof of the second part is entirely similar. It is sufficient to prove the lemma form < oo.
From Lemma 4.1, we see that our assertion is a consequence of the following:
If h E cm-l,A(V), h = 0 on V\D, and qJ E cm- 2((V\D) X V), then
J(r) =f. h(z )ffJ(f, z ){fk- zk) dz 1\ dz E cm-2{V\D). {4.2) D, ~r+ I I r - z 12(n-r-J)
For r E V\D, (4.2) is a proper integral, so we may differentiate under the integral sign m - 2 times. Hence I(n E cm-2(V\D).
The derivatives of the integrand are linear combinations of expressions of the form
- h(z)Dr.T'_l_. DJ,J' fk - zk (r z) l/1- r ~r+l r lr-zl2<n-r-I)X • •
where III+ I'+ J + J'll < m- 2 and x(r, z) is continuous on (V\D) X V. And we have
I D/,1' I I.;;; Ml
r (~(r,z))'+l l~(r,z)lnr+rn+r+l'
DJ,J' fk - ik .;;;; M 1
r I'- z 12(n-r-1) I'- z 12(n-r-I)+IJJ+J'II-1'
where M1 does not depend on (r, z) E (V\D) X i5. We have h = 0 on aD= {z: p(z) = 0}; hence by Hadamard's lemma
(Amol'd (1], §2.6.4) there exists h1 E cm-4,A(Y) such that h = ph 1; in tum, we have h1 = hp-1 = 0 on V\D, and so, by continuity, h1 = 0 on aD. Repeating this procedure m- 3 times, we get h = pm-3h2 , where h2 E C0•A(V) and h2 = 0 on V\D. For>..' such that 0 < >..'<>..,the function h0 = h2 ·If- z I-A' lies in C((V\D) X D); hence
.;;;; M21 p( Z) lm-3+A' lvl I ~(r, z) 1ur+rn+r+l I r _ z l2<n-r-I)+IIJ+J'U-1
.;;;; I ~~~ ~) r f+/'11 .,, ;~ ~ ,,IIJ + J'll- 1 . I ~a' z) r+ I ·I~~ z 12n-A'-2(r+ I) '
where M2 and M3 are suitable constants. Now, from property 4) of P(f, z) and the fact that p(z).;;;; 0 for z E i5 and
p(n > 0 for f E V\D, we have
1 lp(z >I< c. l~<r. z >I. 2 1
lr- zl ,...; c21~(r. z)l
and, further, I p(z) I< C31 r- z I for z E i5 and' E V\D. Hence
lvl < I r _ ~2n-A' • where M is a constant. But this means that, for r E V\D, the integral of any derivative of the integrand in (4.2) is convergent uniformly in r. i.e. all
derivatives of J(n of order up tom- 2 extend continuously to V\D. Now ( 4.2) is a simple consequence of the following assertion.
Let D be a bounded domain in Rn with boundary of class em, m ~ 0, and let f E em( D). Suppose f and all its derivatives of order up tom extend continuously to i5. Thenf E em(f)).
For n = 2, the proof of this statement is given in Fikhtengol'ts ((1], Para graph 260). For n > 2 the proof is entirely analogous.
4°. We can now proceed to the solution of the problems formulated in I 0 •
THEOREM 4.3. Let D be a bounded strictly pseudoconvex domain with boundary of class em+l, and let y E zr;,:~>(Cn), m ~ 0. Suppose supp y c D. Then:
I. For q < n, there exists a E C(;,q-t)(Cn) such that supp a C jj and a a = y. II. For q = n, an a E C{;,n-t>(Cn) such that supp a C jj and a a = y exists if
and only if
/, p. 1\ y = 0, p. E An-p(D). (4.3) D
III. If~ E e<;:~-t>(Cn) and aa = y,C) then there exists /3 E e<;~~I)(Cn) such that ap =a on CD, q ~ 2.
PRooF. The necessity of (4.3) is a simple consequence of Stokes' formula. Indeed, for any p. E An-/D). ·
( p. 1\ y = /, d(p. 1\ a) = ( p. 1\ a= 0. )D D )3D
Let us prove I and the remaining part of II. Let V be the domain of holomorphy introduced in 2°. Since supp y C jj and ay = 0, for z E V we have by (1.1)
y{z) = ai},q-t(V, y) = ~l},q-t(D, y). (4.4)
If q = I, then I},q-t(D, y) E Ap(eD), since supp y c i5. By Hartogs' theo rem, I},q-t(D, y) can be extended to all of en as a form a 1 E Ap(Cn). It now follows from (4.4) and Corollary 2.4 that a= Ii.q-t(D, y)- a 1 is a form with the desired properties.
We now consider the case q ;;;;. 1. Then, by Lemmas 1.3 and 1.4, for z E V\D we have
I},q-t(D,y)(n = fv.-r(z) AazP.!-t A p!(ni-p)!Dp,n-p(az,an
- ( I I +a lv,y(z) 1\ P.q-2 1\ p !(n _ p )! Dp,n-p{az, an, (4.5)
(')If q = n, it will be supposed that y satisfies (4.3).
forq < n, and
(4.6)
(for convenience, we have deviated from our usual notation and interchanged the places of r and z ).
Using Stokes' formula and the equality y lao= 0, we see that the first integral in (4.5) vanis~s. Since U(f, z), and consequently ~.0(u, f, z), are holomorphic in z on D, the first integral in (4.6) also vanishes by virtue of (4.3). By Lemmas 1.3 and 1.4 we see that the second integrals in (4.5) and (4.6) are linear combinations of the integrals considered in Lemma 4.2; hence
I},q-l(D, y) = ap on V\D {4.7)
and p E c<~~~2>(V). _ Corollary 2.4 and (4.4) now show that a= f}.q- 1(D, y)- ap is a form with
the desired properties. ' It remains to prove III. By (1.1 ), we can represent a as
-1 () 2 ( -2 a-Ip,q-l V,a -Ip,q-l V,y)-alp,q- 2(V,a).
As in the proof of Theorem 2.1 0, we can show that
I~.q-l(V, a) E Z{;,q-t)(V).
Since Vis a domain of holomorphy,
l;,q-l(V, a)= fJ{J 1
with /11 E C(~.q- 2)(V). The proof is completed by applying (4.7) and Corollary 2.4.
THEOREM 4.4. Let D be a bounded strictly pseudoconoex domain with cm+ 2
boundary andy E zr;,·.~)(i)). m ;;;a. 1. Then there exists a E G~.q-I)(D) such that aa = y.
PRooF. Consider the domain of holomorphy V introduced in 2°, and a y E C(~:~>(C") such that y =yonD and supp y c V. By (Ll),
y = -I},q(V, fJy)- fJI},q-l(V, y ).
For q = n we have I},n(V, fJy) =: 0; hence the theorem follows in this case from Corollary 2.4.
Now let q oe;; n- 1. Then, since ay = 0 on D, Lemma 1.4 yields for z ED
Y(z)
=a{-i a-y(n !\ 1'~-l !\ 1 ( I_ )I Dp,n-p(az, an- I},q-l(V~ y )} (V\D>c p. n p .
-i a-y(n !\ arl'~ !\ 1 ( 1_ )I Dp,n-p(az, an. (V\D>c p. n p .
Since a-y = 0 on a(V\D), the second integral above vanishes by Stokes' theorem. The first integral is a linear combination of integrals considered in Lemma 4.2; hence the proof is by appealing to Lemma 4.2 and Corollary 2.4.
CoROLLARY 4.5. If D is a strictly pseudoconvex domain in C", n ~ 2, with boundary of class cm+l, m ~ 2, and g E cm·"(CD),and ijag can be extended to CD as a form of class C(O.~ 1·"( CD), then g can be extended to C" as a function in cm(C").
Also, for every extension of a8 !_O a form a E Zt0;1~(C"), there exists an extension/ E cm(C") of g such that aj= a.
PROOF. Suppose a! is ~tended to a1 E C(O,~ 1·"(C"). Th_:n aa1 E C<O,i) 2·"(C"), and supp aa1 c D. Further, if n = 2, for any I' E A2(D) we have
/.~& !\ aat =f. I'!\ aa. = 1 I'!\ al = 1 I'!\ ag = 0. o D1 ao1 ao1
Here D 1 is a domain with smooth boundary such that D u; D 1 and I' E A2(Dt). Now Theorem 4.3, III shows that there exists h E cm(C") such that ah = at
on CD. The function cp = h- g is holomorphic on CD; hence by Hartogs' theorem it can be extended to an entire function. Therefore g = h - cp can be extended to C" as a function of class cm(C").
To prove the second part of Corollary 4.5, we note that, by Theorem 4.4, a= af. with It E cm(C"). Now g- It is holomorphic in CD; let /2 be the entire function extending it to C". Then f = / 1 + /2 is the desired extension.
This corollary is a generalization of Hartogs' theorem to smooth functions. For domains which are not domains of holomorphy, an assertion similar to Corollary 2.5 is false. An example of a function g not extendible continuously to CD, for which ag admits a c~ extension to CD, will be constructed in §8 (see (8.1)).
CHAPTER II
§5. Polynomials orthogonal to holomorphic functions
1 o. Let D be a bounded domain in C", with smooth boundary, and let 0 ED. Let wk(z), k = I, ... ,n, be C1 functions on aD such that, for all z E aD,
(w(z),z)= I'. (5.I)
w(w, z) = ~ (-I)k-lwkdw[k] 1\ dz, k=J
where w = (w1, ••• , w,.) and [k] signifies that dwk is to be omitted. If w(w, z) is nondegenerate on aD, then every a E C2,._ 1(aD) can be written as«= IJI(z)w, with cp E C(aD). Instead of discussing forms orthogonal (with respect to integration over aD) to holomorphic functions, we shall discuss functions cp that are orthogonal to holomorphic functions in the sense that
r f(z)cp(z)w(w, z) = 0 lao
(5.2)
for all f E A( D). We shall denote the subspace of such functions cp E qaD) by O(aD).
In certain situations, it is useful to have a description of the polynomials P(z, w) which lie in O(aD). Observe that, for n = I, P(z, w) and w(w, z) are respectively P(z, I/z) and dzjz.
( 1)Note that w(w, z) = w(w, z,O) of §1.2°.
37
THEOREM 5.1. P(z, w) E O(aD) if and only if
1 ( 'TI 'Tn-1 1-T~-···-'Tn-1) p z1, ••• ,z,,- , ... , --, d'TI 1\ · • • 1\d'T,-1.
T1;;o.O, ... ,Tn-l;:o.O Z1 Zn-1 z, Tt+ • • • +Tn-to;;l
" ( 1 1 ) = ~ Qj zj, z1, -, ... , (j), ... ,z,,- , j=l zl z,
(5.3)
where the Q j are polynomials such that Q /0, z 1, ••• , 1j z,) = 0, j = 1, ... , n.
We divide the proof of this theorem into several lemmas.
LEMMA 5.2. Iff E A( D) and P(z, w) has the form m
P(z, w) = ~ aak,pkZ0kW/Jk, k=l
where ak = (af, ... ,a!) and pk = (/Jt, ... ,p:), then
1 ( ·)" ~ [ pko /(z)za• ] P(z, w)f(z)w = 2'1Tl ~ aak,pk D ' ( k ) •
ao k= 1 II P II + n - 1 ! z=O
(5.4)
(5.5)
This lemma follows easily from the Cauchy-Fantappie formula (see (13.2)), written (for r sufficiently close to 0) in the form
/(r) = (n- 1~! 1 /(z)w(w, z)". (2'1Ti) ao(1-(f,w))
LEMMA 5.3. In order that a polynomial P(z, w) of the form (5.4) lie in O(aD}, it is necessary and sufficient that, for all n-tuples y = (y1, ... ,y,) of nonnegative integers,
(5.6)
PRooF. It follows from (5.5) that (5.6) signifies orthogonality of P(z, w) to all monomials z7 ; hence the necessity is obvious. The right side of (5.5) depends neither on the form of D, nor on the concrete choice of the w;(z), i = 1, ... ,n. Thus, whether P(z, w) lies in O(aD) or not depends only on the properties of P(z, w) itself. In particular, P(z, w) E O(aD) if and only if P(z, w) E O(aB(O, r)), whereB(O, r) = {z: I z1 12 +···+I z, 12 < r 2}. We have seen in Examples 1 and 2 of 3° how thew;(z) can be conveniently chosen for a ball. Each f E A ( B( 0, r)) can be uniformly approximated on aB(O, r) by po1ynOinials in z. Hence the orthogonality of P(z, w) to all monomials z7
already implies that P(z, w) E O(aB(O, r)).
§5. POLYNOMIALS ORTHOGONAL TO HOLOMORPHIC FUNCTIONS 39
Finally, from (5.6) we deduce
LEMMA 5.4. A polynomial P(z, w) of the form (5.4) lies in O(oD) if and only if, for all n-tuples y of nonnegative integers,
pk I ... f3k I ~ ak k I' n· =0
pk-ak=y a ,p (II.Bkll + n- 1}! . (5.7)
The proof of Theorem 5.1 now reduces to comparing (5.7) with the equality
1 ( T1 Tn-1 1 - Tl- ''' -'Tn-1) P Z, -, ... , --, dTI 1\ · · · 1\d-rn-1
T1;;.0, ... ,T._ 1:>0 Z1 Zn-1 zn
= ~ L._:.~:····· (II::~:~:~\)}'· CoROLLARY 5.5. Every polynomial P(z, w) can be represented as the sum of a
polynomial Q(z, w) E O(oD) and a polynomial R(w) depending only on w.
PROOF. It suffices to consider a polynomial P(z, w) of the form (5.4) for which .Bk - ak =· y is a constant vector. If' at least one coordinate of y is negative, then P E O(oD). If all the coordinates of yare nonnegative and (5.7) is satisfied, then again P E O(oD). It remains to consider the case when all the components of yare nonnegative and (5.7) does not hold. In this case, we can add a monomial a0,..,w.., to P, choosing a0,.., so that (5.7) is satisfied for the polynomial P + a0_..,w..,, i.e. that this "augmented" polynomial lies in O(oD).
Note the inclusions A0(oD) c O(oD) c C(oD), where A0(oD) is the space of traces on oD of functions f( z) holomorphic in D and continuous on i5 with f(O) = 0. From Theorem 5.1 it follows that O(oD) is not a ring for n > 1. We stress that the characterization of the polynomials in O(oD) given by Theorem 5.1 and the representation in Corollary 5.5 do not depend on the concrete form of D.
2°. For large classes of domains, the functions w1(z), ... , wn(z) can be chosen so that
polynomials P( z, w) are dense in C( oD). (5.8)
Then the forms P(z, w)w are dense in C2n_ 1(oD). The following theorem may be considered as a new step in the characterization of forms orthogonal to holomorphic functions.
THEOREM 5.6. Ifw; E C2(oD}, i = 1, ... ,n, then a polynomial P(z, w) belongs to O(oD) if and only if the form Pw can be extended into D as a a-exact form of type (n, n - 1).
40 II. FORMS ORTI:IOGONAL TO HOLOMORPHIC FORMS
For the proof, we need
LEMMA 5.7. P(z, w) E 0(3D) if and only if P(z, w) can be represented as a linear combination of polynomials of the following two kinds.
zawfl, with ak > Pkfor some k,
ZaWa+-y( Ca+-y.l- ZtWI);
(5.9}
(5.10)
here a, f3 andy are nonnegative integral vectors, and Cp.t = ({J1 + 1)(11/JII + nt1•
PROOF. It foilows from Lemma 5.4 that polynomials of the form (5.9) or (5.10) lie in 0(3D). To prove the necessity, it is enough in view of the same lemma, to show that a polynomial
m w"Y ""' a kZ 0 kW 0 k
~ a ' (5.1 I} k=l
lying in 0(3D) is a linear combination of polynomials of the form (5.10). Without loss of generality, we may suppose that a 1 = 0 and a~ ¥= 0,
k = 2, ... ,m. Then(5.ll)can be written as
(5.12)
The first term in (5.12) is a linear combination of the polynomials (5.10); the remaining polynomial has lower degree in z than the initial one and also lies in 0(3D). Repeating this argument, we wiii have (after finitely many steps) a representation of (5.11) as a sum of a linear combination of the polynomials (5.10) and a polynomial bw"~'; by Lemma 5.4, bw"~' E 0(3D) only if b = 0.
PRooF OF THEOREM 5.6. Sufficiency follows immediately from Stokes' theorem. We shaii prove the necessity for polynomials (5.9) and (5.10) (see Lemma5.7).
Suppose the polynomial has the form (5.9). We may suppose that k = 1. Then
zawfl = (z1w1)fl1Zi1 • • • z:•wfz · · · wf•zj•-ft•
= Z01 -fl1(l- z2w2- ... -z w )fl1Z01 ••• Z0 •wflz .•. wfl· I nn 2 n 2 n•
and so it is enough to consider the polynomial P = zj• · · · z:•wfz · · · wf• with a 1 >0.
By Lemma 1.6, for zk =F 0 we have
w(w, z) = (n- I~! (-l)k-l_!_dw[k] /1. dz. (5.13) (27Ti) zk
Therefore, for z 1 =F 0
Pw = (n- I)! z«•- 1z«z · · · z«•w2flz · · · wfl•dw[l] /1. dz (27Ti)" I 2 n n
=a{ (n- I)! z«•- 1z«z. · ·z .. •wflz+l · · · wfl•dw[l,2] Adz}. (27Ti)"(/J2 + I) I 2 n 2 n
This equality can be extended to all of aD by continuity. It remains to consider the case when P E O(aD) has the form (5.10). In this
case
For z1 • • • • • zn =F 0 we get from (5.13)
(n- I)' n Pw = · z«w«+y ~ (-l)m-lamwmdw[m] /1. dz.
(27Ti)" m=l (5.14)
The condition z1 • • • • • zn =F 0 may be dropped since both sides of (5.14) are continuous on aD. If m > I, then
a( wlwmz«w«+ydw[l, m] /1. dz)
= (a1 + y1 + l)z«w«+ywmdw[m] /1. dz
+ (-l)m(am + 'Ym + l)z«w«+Yw1dw[l] Adz,
and so (5.14) implies
where a is a constant, and p. a linear combination of the forms w1wmz«w«+Ydw[l, m] 1\ dz. Further, in view of (5.13),
(27Ti)" - - Pw = az w z«w«+flw + a = P w + a (n _ I)! 1 1 ,. 1 ,..
Hence P 1(z, w) E O(aD), and it follows from (5.7) that a = 0. 3°. We now consider examples of classes of domains for which there exists a
vector-valued function w(z), z E aD, such that (5.1) and (5.8) hold and the form w(w, z) is nondegenerate.
42 II. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS
EXAMPLE I. Let D = {z: p(z) < 0} be a strictly pseudoconvex bounded domain with C 2 boundary and containing zero; suppose that D is also linearly convex (Aizenberg [1]), which in this case means that, for every z E aD, the analytic tangent plane {r: (rl - ZJ)P; + · · · +(rn- zn)P; = 0} does not in-• . tersect D. We set
i = 1, .. . ,n. (5.15)
For this choice of w;. (5.1) holds. Let us consider the form w formed from theW; as in 1°.
LEMMA 5.8. w(w, z) is nondegenerate on aD. PROOF. We represent aD as the union of disjoint sets f 1, ••• ,rn such that
pf =I= 0 on fm. Then we see by an easy computation that, on fm, .. "'= (-1)m- 1e(p )dZ[m] 1\ dz
I ( I + + 1 )n > P;., Z1Pz1 • • • ZnPz.
where e(p) is the generalization of the Levi determinant (see Fuks [1], Chapter II, §12.3, or Vladimirov [1], Chapter Ill, §18.5) to the case of n variables (see Rizza [1]):
0 I
I
Pz.i.
It remains to show that e(p) :F 0 on aD for strictly pseudoconvex domains D. But this can be easily checked by making a nonsingular linear change of variables such that the analytic tangent plane to D at a given point of aD is parallel to a coordinate hyperplane, and using the following property of e(p ): if z = z(n is a biholomorphic map, and p1(n = p(z(n), then
e(pl) = e(p)l a(zJ•···•zn} 12· (5.16} a(rl ..... rn)
The proof of (5.16) in the general case is the same as in the case n = 2 (see Fuks [1], Chapter II, §12.3).
Let h be the locus of (z, w) in C2n as z ranges over aD. As in the proof of Lemma 5.8 (by means of the same linear change of variables), it is easy to prove
LEMMA 5.9. his a smooth manifold without complex tangent vectors.
Let f> = { w: ( w, z) =I= 1 for all z E D} be the compact set dual to D, and let = - = D be the dual of D. Dis an open set (see Aizenberg [1] and [2]). Let us also
require that D be connected.e) Then D is linearly convex in the sense of Martineau [1].
LEMMA5.10.
where h is a polynomially convex compact set.
PROOF. f> may be interpreted as the set of complex direction vectors w of analytic hyperplanes of the form aw = {z: (w, z)= I} that do not meet D. Hence the right side of (5.17) is the set of pairs (z, w) in C2" such that z E aD and aw passes through z but does not meet D. From the uniqueness of the analytic tangent plane, it follows that w has the form (5.15). Thus (5.17) holds. i5 and D are polynomially convex, since f5 and D are connected (see, for example, Makarova, Kudaiberganov and Cherkashin [1]); hence (5.17) implies the polynomial convexity of h.
In view of Lemma 5.9, continuous functions on hare uniformly approxima ble by holomorphic functions (Harvey and Wells [ 1 ]). These in tum can be approximated by polynomials P(z, w), because of Lemma 5.10. Thus condition (5.8) is satisfied.
ExAMPLE 2. Now, in contrast to (5.15), we set
lij=ii/lzl2 , j= l, ... ,n. (5.18)
For this choice of the liJ• the vector-valued function w(z) does not depend on D, and always satisfies (5.1). Let us consider conditions under which (5.8) holds. Let D be a bounded domain with 0 ED. Let B denote the uniformly closed algebra of functions on aD generated by z1,. •• ,z,. and w1,. •• , w,, and let Babe its restriction to the (complex) one-dimensional analytic plane a, 0 Ea. Oearly, B separates points on aD and contains I. Let a= {z: z =at= (a 1t, ... ,a,.t), t E C1}, a E C". Then~= ail al-2r 1, so that Ba is generated by t and r 1•
LEMMA 5.11. For (5.8) to hold, it is necessary and sufficient that, for every a, the set aD n a have no interior points and that its complement in a consist of two connected components.
For the proof, we need the following results of Cirka [1]. Let X be a compact set inC", a subalgebra of C(X) and F = {fa}aer a family
of real-valued functions in C( X). Then a continuous function f on X can be
e > 1t is not clear whether the connectedness or iJ is a consequence of the hypotheses of Example I.
approximated uniformly on X by polynomials in elements of A and F if and only if f can be approximated on every X,.= {x EX: /,.(X)= a,.}, a,. E R, by ele ments of A.
PRo_oF OF LEMMA 5.11. Consider the functions/jk = zjzk I z l-2,j, k = 1, ... ,n,
ljk = /kJ• ljk E B. Then ljk + /kJ = 2 Re ljk E B. These functions do not sep arate points lying in the same plane. By taking the restriction of ljk to two planes a 1 and a 2 and considering the functions ljkljJ-l• it is easy to see that Re /jk and Im /jk separate points of different planes. Thus the set { z E aD: Re ljk = b1k, Im ljk = c1k} is either empty or coincides with aD n a for some a.
Sufficiency. If aD n a satisfies the conditions of the lemma, then, by a theorem of Mergeljan [1], polynomials in t and r 1 are dense in qaD n a). Hence Cirka's result implies that B = C(aD).
Necessity. If B = qaD), then
B,. = c(aD n a). (5.19)
In the interior of aD n a, each function in B,., being a uniform limit of holomorphic functions, is holomorphic; hence (5.19) can hold only if aD n a has no interior points. If the complement of aD n a had a component w containing neither 0 nor oo, then for t0 E w we would have (t- t 0r 1 E
qaD n a); but (t- tor• fiB,. by the maximum modulus principle. It remains to consider the question of the nondegeneracy of the form w for
the choice (5.18) of w(z).
LEMMA 5.12. Let D be a bounded domain with smooth boundary. Then w( w, z) is nondegenerate on aD if and only if, for all z E aD
(5.20)
i.e. no analytic tangent plane passes through 0 ED.
~
w(w,z)=~ ± (-l)k-lzkdi[k]l\dz lz I k=l
= (-~)" (z 1p; + · · · +znp; )di[n] 1\ dz. I z I "pi_ I •
Thus the nondegeneracy of w is equivalent to (5.20). Thus all the required conditions of 1 o and 2° ar~ satisfied by the function
w(z) of (5.18), provided D satisfies the conditions of Lemmas 5.11 and 5.12. ExAMPLE 3. If the class of domains D is the same as in Examples I or 2, but
p E C"+2, then, by Theorem 5.4 of Hormander and Wermer [1], (5.8) is valid
§6. THE CASE OF STRICTLY PSEUDOCONVEX DOMAINS 45
for polynomials in Z and V, where V = (v1, ••• ,Vn), each V; being a en+ I function sufficiently close to W; in the C2 topology, i = I, ... ,n. Thus it remains only to choose the V; such that the analogue of (5.1) holds and w( v, z) is nondegenerate on aD.
EXAMPLE 4. Let D be a domain of the class of Example 1, and let W; E en+ I as in Example 3. We extend the w; to some open neighborhood V of aD so that (5.1) is preserved. Consider homeomorphisms cp, c cn+l of aD onto the boundary aD, c V of a domain D, depending continuously in the C2 topology on the parameter t, 0 < t < 1, such that lim,_o cp, in the C2 topology is the identity map of aD. Again applying Theorem 5.4 of Hormander and Wermer, we see that, for e > 0 sufficiently small, the polynomials P(z, w) have the property (5.8) on aD,, 0 < t <e. Thus thew; of (5.15) work not only for the domains of Example 1, but also for domains which are close to them in the above sense.
§6. Forms orthogonal to holomorphic forms:
the case of strictly pseudoconvex domains
1 o. We now proceed to the characterizatiol) of forms orthogonal to holomor phic functions, i.e. we shall answer the following question:
Let D be a bounded domain with smooth boundary in en. Then which forms a E C(p,n-l)(aD) are orthogonal to functions in An-p(D) in the sense that
11'1\a=O ao
for alii' E An-p(D)? Let us denote the space of these orthogonal forms by A;_P(D). As already observed, in the case n = I, for any domain D, we have
.1. -. • I -a E A 1_p(D) tf and only tf a E Z(p,o)(D) n Cp,o(D),p = 0, 1. It is easy to give a sufficient condition for a form a to belong to A;_P(D).
Indeed, if a = y lao for some y E Z(1p,n-l)( D), then a E A;_P( D), since
11' 1\ a = J. d(p.l\ y) = 0 ao o
by Stokes' theorem, for any I' E An_p(D). By a standard application of the Hahn-Banach theorem, it can be seen from
Theorem 2.1 that Zt~.n.=.•>(D) is weakly dense in the space of forms (and even measures) from ACJ. (D) in the topology of C*(aD), the dual space of qaD).
2°. For strictly pseudoconvex domains, we can show that the condition stated in I 0 is not only sufficient, but also necessary. More precisely, we have
THEOREM 6.1. If D is a strictly pseudoconvex domain in en, n > I, with aD E cm+I.\ m;;;.. I, an_!! if a E c1;~_ 1>(()D), then a E A;_p(D) if and only if there exists y E Cp~n-z(D) such that ()y lao= a.
Theorem 6.1, Part II of Theorem 4.3 and Part III of Theorem 4.3 for q = n are essentially equivalent.
PRooF. Sufficiency was proved in I 0 •
Necessity. By Theorem 2.12,
a = a, lao - azlao• (6.1)
where a 1 E Z[';,·.~-I>(D) and_ a2 E Z[';,·.~-I>(CD). If we extend a2 to a form a2 E c<;:~·-l)(en), then y = ()a2 satisfies the conditions of Theorem 4.3:
f p. 1\ y = f p. 1\ aa2 = 1 p. 1\ az = 0, D D aD
since a2 lao =a- a1 lao• and a, a1 lao E A;_P(D). By Part III of Theorem 4.3, a 2 = a2 = ap on CD, and fJ E c1;,n-z>(en).
By Theorem 4.4, a 1 = ay, and y1 E ~;.n-z>(D). Now (6.1) yields that y = y1 - p has the desired properties.
Theorem 6.1 cannot be carried over to arbitrary domains because it is not true that all a-closed forms (which, as shown in I o, are orthogonal to holomorphic forms) are a-exact for every domain. Thus the following result seems natural:
k - THEOREM 6.2. Let D = 0\ U1 (0;), where 0 and the 01 are strictly pseudo-
convex domains with cm+Z boundary, m;;;.. I, such t!!_at O;CS:: 0 andO; n oj = 0
fori= j; let~ E ~;:~-I>(()D). Then a E A;_p(D) if and only if there exists a E '4.";-:,,I_,>(D) such that ii lao= a.
The "exotic" form of the domain in which such a characterization is given reflects to a certain extent (and for n = 2 almost completely) the true nature of things, as Theorem 8.1 shows.
PRooF. Let a E A;_p(D). By Theorem 2.12, a= a1 lao- a2 lao• where a 1 E Z<";·.~-I>(D) and a 2 E zt;·.~-I>(CD). a 2 breaks up into forms a0 , a1, ••• ,ak, a0 E Z['l;.~-I>(CO), a1 E Z1";·.~-I>(01 ). By Theorem 4.4, a1 = ay1,
Y; E ~;.n-2)(01 ). Extend theY; to en so that k
supp Y; <& 0\ U Oi, j=l j+i
and denote the extended forms also by y1• Consider p = a 2 - (ay, + · · · +ayk). Clearly; P = 0 on 01, i = 1, ... ,k, and p = a0 on CO. Further,
§7. TilE GENERAL CASE 47
P lao E A;_p(D), since ~)y; lao and a 2 lao belong to A;_P(i)). And since 1n-p(U) = An_p(D}, we ha!_e a0 lao E A;_p(li). By Theorem 6.1, a 0 lao= 3y0 lao• where Yo E C(;,n-I)(Sl). Let cp E C00(Cn), cp = 1 on CSl and cp :: 0 on U~ Sl;. Then p lao = 3( cpy0 ) lao· Set
k
a= al- ~ ay;- a{cpyo). i=l
Then a E zr;,-:.."- I)( D), and
a lao= al lao- ( i~l ay;" lao+ a( cpyo) lao) =a.
Fork= 1, Theorem 6.2 admits an alternative formulation:
COROLLARY 6.3. Let D 1 C& D2 be strictly pseudoconvex domains with cm+l boundaries, m ;;;.}, and let a; E C(;:~-l)(aD;), i =_1,2, .... Then the a; are restrictions to the aD; of the same form a E zr;,-:,I_ 1>( D2\ D1) if and only if
for any p. E An-p(i>;).
3°. Just as Parts II and III of Theorem 4.3 for q = n yield Theorem 6.1, Parts I and III for q < n yield
THEOREM 6.4. If Dis a strictly pseudoconvex domain in en, n ;;a. 2, with cm+l boundary, m ;;a. 1, and if ~ E zt;,·.~>(CD}, 1 < q < 11- 2, then there exists y E C(;.q-I)(Cn) such that ay = a on CD.
This theorem is an analogue of Hartogs' theorem on the extension of holomorphic functions from the exterior of a compact set.
§7. The general case
For the characterization of forms orthogonal to holomorphic forms in the general case, we need the notion of the envelope of holomorphy of a closed bounded domain. If a compact set K can be written as nm Dm, Dm+ 1 ~ Dm, where each domain Dm has a schlicht (i.e. univalent) envelope of holomorphy H(Dm}, then we shall say that K has a schlicht envelope of holomorphy, and define H(K) as nm H(Dm). The envelope of holomorphy H(K) so obtained does not depend on the choice of the sequence Dm, m = 1, 2, ... , and preserves a number of properties of the envelopes of holomorphy of domains (see Aizenberg [2]).
48 II. FORMS ORTIIOGONAL TO HOLOMORPHIC FORMS
THEOREM 7.1. If 3D E cm+l,", m ;;as 1, a compact set jj has a schlicht envelope of holomorphy, a'!_d 'Y E C(~:~-J)({}D), then y E A;_P(D) if and only if there exist y1 E zt;·.~-I>(D) andy2 E Bp,n_:l(CD) n C(~:~'- 1>(CD), 0 <"'A'< "'A, where 'Yz = 0(1 z 1'-2") as I z , .... 00 and is a-exact in CH(D), such that
'Y = 'Y

differential forms orthogonal to holomorphic functions or forms, and their properties

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