essential norm of composition operators on the hardy space h1 and the weighted bergman spaces on...

Arch. Math. 98 (2012), 327–340c© 2012 Springer Basel AG

0003-889X/11/040327-14

published online March 30, 2012DOI 10.1007/s00013-012-0367-1 Archiv der Mathematik

Essential norm of composition operators on the Hardy spaceH1 and the weighted Bergman spaces Ap

α on the ball

Stephane Charpentier

Abstract. We estimate the essential norm of a composition operator actingon the Hardy space H1 and the weighted Bergman spaces Ap

α on the unitball. In passing, we recover (and somehow simplify the proof of) parts ofthe recent article by Demazeux, dealing with the same question for H1

of the unit disc. We also estimate the essential norm of a compositionoperator acting on Ap

α in terms of the angular derivatives of φ, under amild condition on φ.

Mathematics Subject Classification (2010). Primary 47B33;Secondary 32A36.

Keywords. Angular derivative, Carleson measure, Composition operator,Essential norm, Hardy space, Several complex variables,Weighted Bergman space.

1. Introduction. Throughout this paper, BN = {z = (z1, . . . zN ) ∈ CN ,

∑Ni=1 |zi|2 < 1}, N ≥ 1, will denote the open unit ball of C

N . For φ : BN → BN

a holomorphic map, we define the composition operator Cφ with symbol φ byCφ(f) = f ◦ φ, for f holomorphic on BN . The continuity and compactness ofthis operator acting on the Hardy spaces Hp(BN ) and the weighted Bergmanspaces Ap

α(BN ), 1 ≤ p < ∞, α > −1, have been characterized in terms of Car-leson measures. In dimension one, the boundedness of Cφ for any φ : D → D isa consequence of the Littlewood subordination principle. In C

N , N > 1, it iswell-known that there exists some map φ : BN → BN such that the associatedcomposition operator is not bounded on Hp(BN ) or on Ap

α(BN ). We refer tothe monograph [5] for a quite complete study of composition operators.

Studying composition operators mainly consists in the comparison betweenthe properties of Cφ and the behavior of its symbol φ. For example, it appearsthat both boundedness and compactness of Cφ on Hp(BN ) and Ap(BN ) are

328 S. Charpentier Arch. Math.

closely related to the way in which φ approaches the boundary of BN . Toget more precise information, it is interesting to compute, or estimate, thenorm and the essential norm of Cφ. We recall that the essential norm ‖T‖e ofan operator T acting on a Banach space E is defined as the distance in theoperator norm from T to the set of compact operators on E1:

‖T‖e = inf{‖T − K‖, K ∈ L (E) compact}.

Cowen and MacCluer [5] contains a lot of information about the norm andsome about the essential norm of composition operators on Hp(BN ) andAp

α(BN ).Before 2011, many authors contributed to the computation or the esti-

mation of the essential norm of Cφ [2,10,15], as well as the essential norm ofweighted composition operators [6,7,9], acting on Hp(BN ) and Ap

α(BN ), alwaysfor p > 1. The main idea consists in approximating the identity operator bya sequence of compact operators, which fits well with the study of composi-tion operators on these spaces. In all these previous articles, this sequence wasalways the same: the sequence of finite rank operators Sn which maps f inHp(BN ) or Ap

α(BN ) to the nth-partial sum of the Taylor series of f. A keypoint was the fact that this sequence is uniformly bounded for p > 1, which iswell-known to be false for p = 1.

More recently [8], Demazeux was interested in completing the study ofthe essential norm of weighted composition operators acting between Hp(D)spaces, and managed to generalize most of the known results to p = 1. To thisend, he considered another sequence of finite rank operators approximatingthe identity on H1(D), namely the sequence (FN )N where FN is the operatorof convolution with the Fejer kernel. His proof relies on Fejer’s theorem anddoes not work in several variables.

The first goal of this paper is to unify, and to some extent to simplify thetreatment of the essential norm of composition operators acting on Hp(BN )and to treat, along the way, the case of H1(BN ), which has always been avoidedin the previous works. We will generalize the results by Cuckovic and Zhao[6,7] to the ball and to any 1 ≤ p < ∞. For the sake of clarity, we will only dealwith composition operators acting on Hardy spaces, but we would like to insiston the fact that there should be no real difficulty to obtain the generalizationsfor weighted composition operators acting between different Hardy spaces, ordifferent weighted Bergman spaces, by combining the ideas of [6] or [8] andthat of the present paper.

In the case of composition operators acting on weighted Bergman spaces,we will continue our investigation of the essential norm in the direction of thelink that exists between the compactness of Cφ and the angular derivatives ofφ (see [1,4,12,18]). For example, in 2001 [18], Zhu proved that if Cφ is boundedon some Ap

β(BN ),−1 < β < α, then Cφ is compact on A2α(BN ) (hence on every

Apα(BN )) if and only if

1 Therefore, ‖T‖e can also be seen as the norm of the equivalence class of T in the Calkinalgebra L (E)/K (E), where K (E) stands for the set of compact operators on E.

Vol. 98 (2012) Essential norm of composition operators 329

lim|z|→1

1 − |z|1 − |φ(z)| = 0. (1.1)

This condition (1.1) is in turn equivalent to the fact that φ does not admitan angular derivatives at any point of SN , according to Julia–Caratheodory’stheorem [14]. Zhu’s result improved that given in [4,12], in which papers theauthors gave a similar result under stronger conditions on φ which alwaysimplies the boundedness of Cφ on A2

α(BN ). Zhu’s proof used a Schur test andthe fact that A2

α(BN ) is a Hilbert space. In 2009, Ueki modified Zhu’s argu-ments to obtain upper estimates on the essential norm of Cφ in terms of thequantity appearing in (1.1) under the assumption that Cφ is bounded fromA2

α(BN ) to A2β(BN ) for some β ≥ α (note that a lower estimate of ‖Cφ‖e in

terms of angular derivatives of φ can be found in [5] for p > 1). Yet, Ueki’sresult is not correct in full generality (see the explanations at the end of thepresent paper). Recently, in [1], we generalized Zhu’s result to the contextof weighted Bergman–Orlicz spaces, using simple computations. In particular,we recovered Zhu’s result for Ap

α(BN ), for any 1 ≤ p < ∞. We will see howa refinement of the proof given in [1] allows to obtain upper estimates of theessential norm of composition operators on Ap

α(BN ) for 1 ≤ p < ∞ in terms oflim sup|z|→1

1−|z|1−|φ(z)| . We will also see that this result is sharp in some sense.

This paper is organized as follows: In the next section, we specify the frame-work and introduce the key point that will allows us to estimate the essentialnorm of a composition operator on Hp(BN ) for any p ≥ 1. The third partconsists in the statement and the proof of this result. The last part is devotedto the essential norm of a composition operator on weighted Bergman spaces.

Notation. Throughout this paper, we will denote by σN the normalized rota-tion-invariant positive Borel measure on the unit sphere SN .vα will standfor the normalized weighted Lebesgue measure on the ball, given by dvα =cα(1 − |z|2)αdv, α > −1, where v stands for the volume Lebesgue measureon BN .

Given two points z, w ∈ CN , the Euclidean inner product of z and w will

be denoted by 〈z, w〉, that is 〈z, w〉 =∑N

i=1 ziwi; the notation | · | will standfor the associated norm, as well as for the modulus of a complex number.

We will use the notations � and � for one-sided estimates up to an abso-lute constant, and the notation ≈ for two-sided estimates up to an absoluteconstant.

When the context is clear, we will write Hp instead of Hp(BN ) and Apα

instead of Apα(BN ).

2. Framework and preliminary results.

2.1. Framework. Let 1 ≤ p < ∞. The Hardy space Hp of the unit ball BN

consists of those functions f holomorphic on BN such that

‖f‖pp := sup

0<r<1

∫

BN

|f(rz)|pdσN (z) < ∞.


Endowed with ‖ · ‖p,Hp is a Banach space. We denote by H∞ the Banach

space of bounded holomorphic functions on BN endowed with the supremumnorm. Fatou’s lemma asserts that for any 1 ≤ p ≤ ∞ and for every f in Hp,there exists f∗ ∈ Lp(SN , σN ) such that fr(ζ) := f(rζ) tends to f∗(ζ) as r → 1,for σN -almost every ζ ∈ SN . Moreover, if 1 ≤ p < ∞, then ‖f‖p = ‖f∗‖Lp ,so that Hp can be viewed as a closed subspace of Lp(SN , σN ). It is also well-known that fr tends to f in Hp as r → 1, whenever 1 ≤ p < ∞. From now, iff ∈ Hp, then we shall denote by the same letter f the boundary function f∗

as well as the function which is equal to f on BN and to f∗σ-a.e on SN .Up to now, Carleson measures have been the main tool used to character-

ize both boundedness and compactness of composition operators on Hp on theball. This involves some geometric notions that we recall here. For ζ ∈ SN and0 < h < 1, let us denote by S (ζ, h) the non-isotropic balls in BN defined by

S (ζ, h) = {z ∈ BN , |1 − 〈z, ζ〉| < h}.

Given a finite positive Borel measure μ on BN , we introduce the function�μ on (0, 1) given by

�μ(h) = supζ∈SN

μ(S (ζ, h))

and the following quantities:

‖μ‖p := sup0<h<1

(�μ(h)hN

)1/p

and ‖μ‖p := lim suph→0

(�μ(h)hN

)1/p

.

By definition, μ is a Carleson measure if ‖μ‖p < ∞ and is a vanishing Car-leson measure if ‖μ‖p = 0. These properties of μ (which do not depend on p)characterize the boundedness and the compactness of the canonical embeddingHp(BN ) ↪→ Lp(BN , μ) respectively (Carleson’s theorem, see e.g. [13]).

For φ : BN → BN , we denote by μφ the pull-back measure of σN under theboundary limit φ∗ of φ: for any E ⊂ BN ,

μφ(E) = σN ((φ∗)−1(E) ∩ SN ).

A classical argument based on a change of measure formula and on the densityof polynomials in any Hp allows us to view the composition operator Cφ onHp as the embedding operator of Hp into Lp(BN , μφ). Therefore, Carleson’stheorem yields [5, Theorem 3.35]:

Theorem 2.1. Let 1 ≤ p < ∞ and φ : BN → BN holomorphic. Then

1. Cφ is bounded on Hp if and only if μφ is a Carleson measure.2. Cφ is compact on Hp if and only if μφ is a vanishing Carleson measure.3. If we denote by ‖Cφ‖ the operator norm of Cφ acting on Hp, then ‖Cφ‖ ≈

‖μφ‖p.

The essential norm of composition operators on Hp(BN ) for 1 < p < ∞has been estimated in such a way (see [2] for example):

Theorem 2.2. Let 1 < p < ∞ and φ : BN → BN holomorphic. Then ‖Cφ‖e ≈‖μφ‖p. If N = 1, this estimate is true when p = 1 [8].


In the literature, the essential norm of composition operators are also esti-mated in terms of integral operators [6,8] for p > 1. We will discuss them morefully later.

2.2. The key point. The sequences (Sn)n and (Fn)n mentioned in the intro-duction are illustrations of the well-known fact that every Hp (1 ≤ p < ∞)has the λ-bounded approximation property. We recall that a Banach space Xhas the λ-bounded approximation property if there exists a sequence (Kn)n

of finite rank operators on X, bounded by λ (we shall say λ-bounded), suchthat for every x ∈ X, ‖Kn(x) − x‖X tends to 0 as n tends to infinity. Yet,when we deal with essential norm of operators, another weaker property maybe sufficient, that is the λ-compact bounded approximation property: we saythat X has the λ-compact bounded approximation property if there exists aλ-bounded sequence (Kn)n of compact operators on X, such that for everyx ∈ X, ‖Kn(x) − x‖X tends to 0 as n tends to infinity. Indeed, we observe thefollowing.

Proposition 2.3. Let λ > 0, let X be a Banach space with the λ-compact approx-imation property and let T ∈ L (X). Let (Kn)n be a λ-bounded sequence ofcompact operators on X, such that ‖Kn(x) − x‖X −−−−→

n→∞ 0 for any x ∈ X. Weput Rn = I − Kn for any n. Then

11 + λ

lim supn→∞

‖RnT‖L (X) ≤ ‖T‖e ≤ lim infn→∞ ‖RnT‖L (X).

Proof. The lower bound follows from the fact that (Rn)n is also (uniformly)bounded (by 1 + λ) and converges pointwise to 0. Therefore, by Ascoli’s theo-rem, it converges uniformly in the strong topology on KBX , for any compactoperator K, where BX is the unit ball of X. Hence

(1 + λ)‖T + K‖L (X) ≥ ‖RnT + RnK‖L (X)

≥ ‖RnT‖L (X) − ‖RnK‖L (X),

as desired, taking the supremum limit for n → ∞ and the infimum over K.For the upper bound, it suffices to use that KnT is compact and write

‖T‖e = ‖RnT + KnT‖e

≤ ‖RnT‖L (X)

so that ‖T‖e ≤ lim infn→∞ ‖RnT‖L (X). �Given a bounded operator T, this observation gives the scheme to compute

or estimate its essential norm: it suffices to find a sequence (Kn)n, as in thestatement of the proposition, which should be convenient enough to handle‖RnT‖L (X). If T is a composition operator acting on Hp, it is very natural tothink about a sequence of compact composition operators. Moreover, it is veryeasy to find examples of such sequences. The most natural one is the sequence(Kn)n of operators from Hp into itself, 1 ≤ p < ∞, defined by Kn = Cφn

forany n ≥ 2, where φn(z) = n−1

n z. Every Kn trivially has a norm less than 1and is compact on every Hp (actually, ‖φn‖∞ < 1 for any n > 1, so Cφn

iscompact on H∞ [5] which implies that Cφn

is compact on any Hp, p ≥ 1).


Moreover, (Kn)n converges pointwise to the identity operator on each Hp,provided 1 ≤ p < ∞. Therefore, if we put Rn = I − Kn for n > 1, thenProposition 2.3 implies that

12

lim supn→∞

‖RnCφ‖ ≤ ‖Cφ‖e ≤ lim infn→∞ ‖RnCφ‖. (2.1)

Remark 2.4. Let (Sn)n and (Fn)n be the sequences mentioned in the introduc-tion. Since (Sn)n is not bounded on H1, because the Riesz transform is notbounded on H1, it can of course not be used to get (2.1) for Cφ acting on H1,with I − Sn instead of Rn. Since Fejer’s theorem is a theorem on the torus T,the sequence (Fn)n cannot of course be used to have (2.1) for Cφ acting on H1

on the ball. Moreover, while ‖(I − Sn)Cφ‖ is not so difficult to compute, whenp > 1, ‖(I − Fn)Cφ‖ is technically very difficult to handle (see [8]).

Since Rn is the difference of two very simple composition operators, thequantities on both sides of Inequality (2.1) will be relatively easy to compute.Let us now turn to the statement and the proof of our main result.

3. Statement and proof of the main result. In this section, φ is a holomorphicself-map of BN and (Kn)n and (Rn)n are the sequences introduced above. Fora ∈ BN , let the test function fa be given by

fa(z) =(

1 − |a|2|1 − 〈z, a〉|2

)N/p

.

Each fa lies in H∞ and ‖fa‖p = 1. We denote by

Bφ,p = lim sup|a|→1

‖fa ◦ φ‖p.

Our main theorem is stated below.

Theorem 3.1. Let 1 ≤ p < ∞ and let φ : BN → BN be holomorphic. If Cφ isbounded on Hp(BN ), then ‖Cφ‖e ≈ ‖μφ‖p ≈ Bφ,p.

Remark 3.2. As already said, the novelty of this theorem is that it is statedfor p ≥ 1 and not only for p > 1.

Proof of Theorem 3.1. Part 1 ‖Cφ‖e � Bφ,p.For any a ∈ BN , since ‖fa‖p = 1, we have

‖RnCφ‖ ≥ ‖RnCφfa‖p ≥ ‖Cφfa‖p − ‖KnCφfa‖p (3.1)

for any n > 1. Since φ(

n−1n BN

)is contained in a compact subset of BN , the

Cauchy–Schwarz inequality yields, for every z ∈ BN ,

KnCφfa(z) =(

1 − |a|2|1 − 〈φ(n−1

n z), a〉|2)N/p

≤ Cn(1 − |a|2)N/p,


for some finite constant Cn independent of z. Therefore ‖KnCφfa‖p −−−−→|a|→1

0

for any n > 1. Taking the supremum limit in (3.1), it follows

‖RnCφ‖ ≥ lim sup|a|→1

‖Cφfa‖p = lim sup|a|→1

‖fa ◦ φ‖p

for any n > 1 and, by Inequality (2.1),

‖Cφ‖e ≥ 12

lim sup|a|→1

‖fa ◦ φ‖p,

as required.

Part 2 ‖μφ‖p � Bφ,p.This part is standard for any p ≥ 1. For the sake of completeness, we prefer

to give the details. It is sufficient to prove that for any a ∈ BN\{0} we have

�φ(1 − |a|)(1 − |a|)N

≤ 4N‖fa ◦ φ‖pp (3.2)

(for the definition of �φ, see Section 2.1). Now, using that |fa(z)|≥ 14N/p(1−|a|)N/p

for z ∈ S (ζ, 1 − |a|) with ζ = a|a| , we have

‖fa‖pLp(BN ,μφ) ≥

∫

S (ζ,1−|a|)

|fa(z)|pdμφ

≥ μφ(S (ζ, 1 − |a|))4N (1 − |a|)N

,

which completes the proof.

Part 3 ‖Cφ‖e � ‖μφ‖p.According to (2.1), we aim to estimate ‖RnCφ‖. Let f ∈ Hp with ‖f‖ ≤ 1

and 0 < r < 1 whose value will be fixed later. We compute∫

SN

|Rn(f ◦ φ)|pdσN =∫

rBN

|Rn(f)|pdμφ +∫

BN \rBN

|Rn(f)|pdμφ

:= I1,n(r) + I2,n(r). (3.3)

By Cauchy integral formula and by Zhu [17, Theorem 4.17], we have

|Rn(f)(z)| ≤ 1n

supw∈rBN

|f ′(w)|

� 1n

supw∈( 1+r

2 )BN

|f(w)|

� 2n(1 − r)N/p

‖f‖p

≤ 2n(1 − r)N/p

for any z in rBN . Therefore, for r fixed, lim infn→∞ I1,n(r) = 0.


Let us now pay attention to I2,n(r), with r fixed. We denote by μφ,r therestriction of μφ to BN\rBN ; since Cφ is bounded by assumption, μφ is a Car-leson measure (see Theorem 2.1, part (1)), and μφ,r is also a Carleson measure.By Theorem 2.1, part (3),

‖Rn(f)‖Lp(BN ,μφ,r) � ‖μφ,r‖p‖Rn(f)‖p � ‖μφ,r‖p. (3.4)

This means that I2,n(r) ≤ ‖μφ,r‖pp for every n.

Now, we denote

Nr,φ,p := suph′≤1−r

(

supζ∈SN

(μφ(S (ζ, h′))

h′N

)1/p)

, (3.5)

and we intend to show that

‖μφ,r‖p = sup0<h<1

(

supζ∈SN

(μφ,r (S (ζ, h))

hN

)1/p)

� Nr,φ,p, (3.6)

where the symbol � involves some constant independent of r. Until the endof the proof, we simply denote Nr,φ,p by Nr. For ζ ∈ SN , let S (ζ, h) be anon-isotropic ball. If h ≤ 1 − r, it is obvious that

μφ,r(S (ζ, h)) = μφ(S (ζ, h)).

Let us assume that h > 1 − r and set γ = h/(1 − r). Let C(h, r) denotethe minimum number of non-isotropic balls of radius 1 − r necessary to coverS(ζ, h). It is known that there exists a constant C ≥ 1, independent of r, hand ζ, such that C(h, r) ≤ CγN (see [14] or [17]). Thus

μφ,r(S (ζ, h)) ≤ CγN supζ∈SN

(μφ,r(S (ζ, 1 − r)))

= CγN supζ∈SN

(μφ(S (ζ, 1 − r))).

By definition supζ∈SN(μφ(S (ζ, 1 − r))) ≤ Np

r (1 − r)N so

μφ,r(S (ζ, h)) ≤ C

(h

1 − r

)N

Npr (1 − r)N ;

Hence

μφ,r(S (ζ, h))hN

≤ CNpr

for any h > 1 − r, which completes the proof of (3.6).By (3.4), we get that I2,n(r) � Np

r for any n. By (3.3) and (2.1), it followsthat

‖Cφ‖e ≤ lim infn→∞ ‖RnCφ‖ ≤ lim inf

n→∞ (I1,n(r) + I2,n(r))1/p � Nr; (3.7)

we finish the proof by letting r tend to 1. �


4. Essential norm of composition operators on weighted Bergman spaces.Throughout this section, φ : BN → BN is a holomorphic map. Let α > −1, 1 ≤p < ∞, and let us denote N [α] = N + α + 1. The weighted Bergman spaceAp

α on the ball is the set of those holomorphic functions f on BN such that‖f‖p

α,p :=∫

BN|f |pdvα < ∞ (see the notations at the end of the introduction).

Every Apα is a Banach space such that fr tends to f in Ap

α as r goes to 1.As for Hardy spaces, the boundedness and compactness of Cφ on Ap

α arecharacterized in terms of Carleson measures. For ζ ∈ SN and 0 < h < 1, wedefine

S(ζ, h) = {z ∈ BN , |1 − 〈z, ζ〉| < h}.

For μ a finite positive Borel measure on BN , we define the quantities

• ‖μ‖α,p := sup0<h<1

(supζ∈SN

μ(S(ζ,h))

hN+α+1

)1/p

;

• ‖μ‖α,p := lim suph→0

(supζ∈SN

μ(S(ζ,h))

hN+α+1

)1/p

.

It turns out that the embedding operator Apα ↪→ Lp(BN , μ) is bounded

(respectively compact) if and only if μ is an α-Bergman–Carleson measure(resp. a vanishing α-Bergman–Carleson measure), i.e. if and only if ‖μ‖α,p < ∞(resp. ‖μ‖α,p = 0) (see [3,11], for example). This leads to characterizations ofthe boundedness and compactness of composition operators on Ap

α, by consid-ering the measure μφ,α defined by

μφ,α(E) = vα(φ−1(E))

for any E ⊂ BN :

Theorem 4.1. (Theorem 3.37 of [5]) Let α > −1, 1 ≤ p < ∞ and φ : BN → BN

holomorphic. Then

1. Cφ is bounded on Apα if and only if μφ,α is an α-Bergman–Carleson mea-

sure.2. Cφ is compact on Hp if and only if μφ,α is a vanishing α-Bergman–Carleson

measure.3. If we denote by ‖Cφ‖α the operator norm of Cφ acting on Ap

α, then‖Cφ‖α,p ≈ ‖μφ,α‖α,p.

We introduce the test functions fa,α, a ∈ BN , defined by

fa,α(z) =(

1 − |a|2|1 − 〈z, a〉|2

)N [α]/p

for z ∈ BN (in order to simplify the notations, we prefer to write fa,α insteadof fa,α,p as well as ‖Cφ‖α instead of ‖Cφ‖α,p). Then fa,α lays in H∞ and‖fa,α‖α,p = 1. We denote by

Bφ,α,p = lim sup|a|→1

‖fa,α ◦ φ‖α,p.

The sequences (Kn)n and (Rn)n introduced in Section 2 still satisfy theassumptions of Proposition 2.3 in Ap

α. The proof of this proposition and that of


Theorem 3.1 can be adapted, without difficulty, to get the following theoremfor weighted Bergman spaces:

Theorem 4.2. Let α > −1, 1 ≤ p < ∞, and let φ : BN → BN be holomorphic.If Cφ is bounded on Ap

α, then ‖Cφ‖α,e ≈ ‖μφ,α‖α,p ≈ Bφ,α,p (where ‖Cφ‖α,e

stands for the essential norm of Cφ acting on Apα).

Remark 4.3. As for Hardy spaces, the novelty of this theorem is that it holdsfor p ≥ 1 and not only for p > 1.

We are now interested in estimating ‖Cφ‖α,e in terms of the quantityappearing in (1.1). For −1 < β < α, we define

Aφ,α,β,p = lim sup|a|→1

(1 − |a|

1 − |φ(a)|)(α−β)/p

.

The following theorem is the main result of this section:

Theorem 4.4. Let α > −1, 1 ≤ p < ∞ and let φ : BN → BN be holomorphic.If we assume that Cφ is bounded on Ap

β for some −1 < β < α, then

‖Cφ‖α,e �(

1 + |φ(0)|1 − |φ(0)|

)N [2α−β]/p

Aφ,α,β,p. (4.1)

For the proof of the previous theorem, we need two results. For 0 ≤ r < 1,we define

Ar,φ = sup|z|≥r

(1 − |z|)(α−β)/p

(1 − |φ(z)|)(α−β)/pand Nr,φ := sup

h′≤1−r

(

supζ∈SN

(μφ(S(ζ, h′))

h′N(α))1/p

)

.

We shall need the following proposition:

Proposition 4.5. Let −1 < β < α, 1 ≤ p < ∞ and let φ : BN → BN be holo-morphic such that φ(0) = 0. We assume that Cφ is bounded on Ap

β(BN ). ThenNr,φ ≤ 2(α−β)/pAr,φ for any 0 ≤ r < 1.

Proof. Let ζ ∈ SN and 0 < h ≤ 1. We compute

μφ,α(S(ζ, h)) =∫

φ−1(S(ζ,h))

(1 − |z|2)αdv(z)

≤ 2α−β supz∈φ−1(S(ζ,h))

(1 − |z|)α−βμφ,β(S(ζ, h))

≤ 2α−β supz∈φ−1(S(ζ,h))

(1 − |z|)α−βhN [β], (4.2)

since Cφ is bounded on Apβ(BN ). Now, using α − β > 0, we obtain

(1 − |z|)α−β

hα−β≤ Ap

r,φ,

for any z ∈ φ−1(S(ζ, h)) and for any h such that h ≤ 1 − r, since 1 − |z| ≤1 − |φ(z)| ≤ |1 − 〈φ(z), ζ〉| < h according to the Schwarz lemma. Thus


μφ,α(S(ζ, h))hN(α)

≤ 2α−βApr,φ, (4.3)

for any h ≤ 1 − r, which completes the proof. �

For a ∈ BN , let ϕa be an automorphism of BN which interchanges a and0.Cϕa

is invertible, and an easy change of variables formula ([17, Proposition1.13]) yields

‖Cϕa‖α ≤

(1 + |a|1 − |a|

)N [α]/p

.

The next lemma will also be useful:

Lemma 4.6. Let a ∈ BN and 0 ≤ r < 1. With the above notations, we have

Ar,ϕa◦φ ≤(

2.1 + |a|1 − |a|

)(α−β)/p

Ar,φ. (4.4)

Proof. Let 0 ≤ r < 1 and z ∈ BN , |z| ≥ r. By Rudin [14, Theorem 2.2.2],

1 − |ϕa ◦ φ(z)|2 =(1 − |a|2)(1 − |φ(z)|2)

|1 − 〈a, φ(z)〉|2so that

1 − |z|1 − |ϕa ◦ φ(z)| ≤ 2

1 − |z|1 − |ϕa ◦ φ(z)|2

= 2(1 − |z|)|1 − 〈a, φ(z)〉|2(1 − |a|2)(1 − |φ(z)|2)

≤ 2(

1 + |a|1 − |a|

)(1 − |z|)

(1 − |φ(z)|) .

Hence Ar,ϕa◦φ ≤(2. 1+|a|

1−|a|)(α−β)/p

Ar,φ, since α > β. �

We are now ready to prove the main theorem of this section:

Proof of Theorem 4.4. We assume that Cφ is bounded on Apβ for some −1 <

β < α. First of all, let a = φ(0) and let ϕa be an automorphism of BN whichinterchanges a and 0. Then ϕa ◦ φ(0) = 0 and Cφ = Cϕa◦φCϕa

because ϕa isan involution. We have

‖Cφ‖α,e = infK∈K

‖Cϕa◦φCϕa− K‖α

= infK∈K

‖(Cϕa◦φ − KCϕa)Cϕa

‖α

≤ ‖Cϕa‖α inf

K∈K‖Cϕa◦φ − K‖α

= ‖Cϕa‖α‖Cϕa◦φ‖α,e.

Cϕa◦φ = CφCϕais bounded on Ap

β(BN ) because Cφ is bounded on Apβ(BN ) by

hypothesis; moreover ϕa ◦ φ(0) = 0. Thus Proposition 4.5 applied to ϕa ◦ φensures that Nr,ϕa◦φ � Ar,ϕa◦φ for any 0 ≤ r < 1. Now, it is not difficult to


adapt the proof of Inequality (3.7) to get ‖Cϕa◦φ‖α,e � Nr,ϕa◦φ, which in turnyields

‖Cφ‖α,e � ‖Cϕa‖αAr,ϕa◦φ. (4.5)

Therefore, Lemma 4.6 ensures

‖Cφ‖α,e �(

1 + |φ(0)|1 − |φ(0)|

)N [2α−β]/p

Ar,φ,

which gives the conclusion letting r → 1. �Now, [5, Theorem 3.43] (which requires 1 < p < ∞) together with Theo-

rem 4.4, gives the following corollary. The last assertion follows from the factthat both boundedness and compactness of composition operators on weightedBergman spaces do not depend on p.

Corollary 4.7. Let α > −1, 1 < p < ∞ and let φ : BN → BN be holomorphic.If Cφ is bounded on Ap

β for some −1 < β < α, then

lim sup|a|→1

(1 − |a|

1 − |φ(a)|)N [α]/p

≤ ‖Cφ‖α,e (4.6)

�(

1 + |φ(0)|1 − |φ(0)|

)N [2α−β]/p

lim sup|a|→1

(1 − |a|

1 − |φ(a)|)(α−β)/p

. (4.7)

In particular, if Cφ is bounded on Apβ for some −1 < β < α, then Cφ is

compact on Apα, 1 ≤ p < ∞, if and only if

lim sup|a|→1

1 − |a|1 − |φ(a)| = 0. (4.8)

Remark 4.8. (1) The last assertion in nothing but Zhu’s result [18] (sinceboth boundedness and compactness of Cφ on Ap

α do not depend on p).(2) In dimension 1, since every composition operator is bounded on any

Apα(D), the compactness of Cφ is equivalent to Condition (4.8), without

boundedness assumption any more.

The second remark above leads us to investigate the case N > 1 more care-fully. Let us mention that the lower estimate in the previous corollary does notrequire the boundedness assumption on Cφ. To be precise, the compactnessof Cφ on some Ap

α, 1 ≤ p < ∞, always implies Condition (4.8). Therefore wededuce the following corollary, which we found nowhere and which seems tous of interest:

Corollary 4.9. Let α > −1, 1 ≤ p < ∞ and let φ : BN → BN be holomorphic.We assume that there exists −1 < β < α such that Cφ is bounded on Ap

β . Ifthere exists γ > −1 such that Cφ is compact on Ap

γ , then Cφ is compact onAp

α.

A natural question is the following: in Zhu’s result, is it possible to replacethe assumption “Cφ bounded on Ap

β for one β,−1 < β < α” by “Cφ boundedon Ap

α”? Actually, the answer is no and MacCluer and Shapiro [12, Theorem6.8 or Corollary 6.9] provided a counter-example:


Theorem 4.10. Let 1 ≤ p < ∞. If N > 1, then there exists α0 > −1 andφ0 : BN → BN holomorphic such that Cφ0 is bounded on Ap

α(BN ) if and onlyif α ≥ α0, and is compact on Ap

α(BN ) if and only if α > α0. The map φ0 canbe chosen with an angular derivative at no point of SN .

This result shows the surprising contrast between N > 1 and N = 1 inwhich context it is known that both boundedness and compactness of Cφ onAp

α(D) do not depend on α.If we consider the limit case β = α in Theorem 4.4, our upper estimate

would become

‖Cφ‖α,e �(

1 + |φ(0)|1 − |φ(0)|

)N [α]/p

,

which inequality contains no information concerning the link between the com-pactness of Cφ and the existence of angular derivatives for φ at any point ofSN . We hope that this may point out in which way the assumption of bound-edness of Cφ on some smaller Ap

β is “necessary” to be able to say that Cφ

is compact on Apα if and only if φ admits angular derivatives at no point of

SN .α = β appears as a case in which every situation can occur.We finish this paper by showing how the previous discussion, especially

Theorem 4.10, indicates that Ueki’s result [16, Theorem 1] is not correct infull generality. Let us quote his result here:

Theorem 4.11. (Theorem 1 of [16]) Let α > −1 and β ≥ α. Suppose that φ isa holomorphic self-map of BN such that Cφ : A2

α → A2β is bounded. Then there

exists a positive constant C such that

lim sup|z|→1

(1 − |z|2)N [β]

(1 − |φ(z)|2)N [α]≤ ‖Cφ‖α,β,e ≤ C lim sup

|z|→1

(1 − |z|2)N [β]

(1 − |φ(z)|2)N [α].

(Here ‖Cφ‖α,β,e is the essential norm of Cφ as an operator acting between A2α

and A2β .)

If this result were true, by taking α = β, it would imply that a compositionoperator which is bounded on A2

α is compact on A2α if and only if φ admits

an angular derivative at no point of SN ([16, Corollary 4]). Yet, this is incontradiction with Theorem 4.10.

Acknowledgement. I am very grateful to the referees for their many valuablecomments and suggestions which improved the paper.

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Stephane CharpentierLaboratoire Paul Painleve, UMR CNRS 8524, Universite Lille 1,59655 Villeneuve d’Ascq Cedex, Francee-mail: [email protected]

Received: 4 November 2011

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