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Hankel Operators on Hardy Spaces

A l i ne Bona m i , M a rc o M . P e l o s o ,

& F r é d é r i c S y m e s a k

In this paper we study the action of certain integral operators on spaces of holo- morphic functions on some domains inCn. These integral operators are defined by using powers of the Szegö kernel as integral kernel. We show that they act like differential operators, or like pseudo-differential operators of not necessarily in- tegral order. These operators may be used to give equivalent norms for the Besov spacesBp of holomorphic functions. As a consequence we prove that, when 1≤ p < ∞, the small Hankel operators hf on Hardy and weighted Bergman spaces are in the Schatten classSpif and only if the symbolf belongs to Bp.

The type of domains we deal with are the smoothly bounded strictly pseudo- convex domains inCnand a class of complex ellipsoids inCn. Our results for strictly pseudo-convex domains depend on Fefferman’s expansion of the Szegö kernel. In this case, its powers act like a power of the derivation in the normal di- rection. The ellipsoids we consider are the simplest examples of domains of finite type. In this case, the symmetries of the domains can be exploited to use methods of harmonic analysis and describe the pseudo-differential operators involved.

1. Basic Notation and Statement of the Main Results

LetD = { z : ρ(z) < 0 } be a smoothly bounded domain in Cn, with ρ ∈ C( ¯D) and∇ρ 6= 0 on ∂D. For p > 0 let Hp(D) denote the Hardy space of holomorphic functions onD, with norm given by

kf kpHp(D) := sup

ε>0

Z

δ(w)=ε|f(w)|pdσ(w),

whereδ(w) := −ρ(w) is equivalent to the distance to the boundary and dσ is the surface measure. LetPS(σ)andSσdenote the Szegö projection and the Szegö ker- nel respectively:

PS(σ)g(z) :=

Z

∂DSσ(z, ζ)g(ζ) dσ(ζ)

forg ∈ L2(∂D). We are interested in the (small) Hankel operator hf(σ)with symbol f, defined for g ∈ L2(∂D) as

Received December 1, 1997. Revision received June 18, 1998.

Michigan Math. J. 46 (1999).

225

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hf(σ)(g) := PS(σ)¡

fPS(σ)(g)

. (1)

Moreover, for p > 0, we define the Besov spaces of holomorphic functions Bpp,n/p(D) by setting

Bpp,n/p(D) :=



g ∈ Lp(D) : Z

Dllg|pB(z, z) dV(z) < ∞

 ,

whereB(z, w) denotes the Bergman kernel on D, dV is the Lebesgue measure onD, and l is some integer such that lp > 2. For simplicity of notation, we shall writeBpforBpp,n/p(D). We also set

kgkBp := kgkLp+

 Z

Dllg|pB(z, z) dV(z)

1/p

.

WhenD is the unit ball in Cn, the Szegö kernel S is known explicitly and S(z, ζ) = cn(1 − z · ¯ζ)−n. In this case, it is quite often useful to study the ac- tion of integral operators defined by using powers of the Szegö kernel. The action of these operators is usually expressed in terms of Besov norms, and it is easily understood via the relation

 I +N

n



S(z, ζ) = S(n+1)/n(z, ζ), (2) whereN is the differential operatorPn

j=1zjzj. Thus, equation (2) shows a link between powers of the Szegö kernel and differential operators.

Using identity (2) and iterations of it, Feldman and Rochberg [FR] proved that in the case of the unit ball, when 1≤ p < ∞, the Hankel operator hf belongs to the Schatten classSpif and only iff is in Bp. Recall that for p > 0, given a com- pact operatorT on a Hilbert space, we say that T belongs to the Schatten class Sp ifP

jsjp< ∞, where

sj := { inf kT − Ek : rank E ≤ j }.

Such results generalized the now classical results of Peller [Pe] and Coifman and Rochberg [CR] for the unit disc. It is natural to ask whether these results are valid in a more general setting.

In order to present the results in this paper we need to introduce some more no- tation. We begin with the case of the ellipsoids.

Letq be a positive integer, and let q be the ellipsoid inC2given by

q := { z ∈ C2:|z1|2+ |z2|2q< 1}.

On these ellipsoids, we first replace the surface measuredσ by the measure dµ defined next, for which explicit computations are available. Precisely, letdµ be the unique measure on∂qsuch that, for positiveF,

Z

C2F(z) dV(z) = Z

0

Z

∂q

F(rz1, r1/qz2) dµ(z)r1+2/qdr. (3)

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The measuredµ is equivalent to the surface measure dσ, and the density dµ/dσ is aCstrictly positive function. We shall denote byPSµandSµthe Szegö pro- jection and the Szegö kernel related to the measuredµ, respectively. The Hankel operatorhf(µ)is defined as in (1), usingPSµinstead ofPSσ. Notice that, in the case of the unit ball, the measuresdµ and dσ coincide.

The identity (2) is no longer valid in the case of ellipsoids, even in an approxi- mate way. ButSµis known explicitly, and we shall prove an identity for the powers ofSµ(z, ζ) that involves some kind of pseudo-differential operators. Specifically, we prove the following results.

Theorem 1.1. Let p > 0 and let λ be a real number such that λ > 1/p. Then there existsc = c(p, λ) > 0 such that, for all g ∈ Bp, we have

1

ckgkpBp ≤ Z

q

Z

∂q

Sµλ+1(z, ζ)

Sµλ(z, z) g(ζ) dµ(ζ)

pB(z, z) dV(z) ≤ ckgkpBp. Notice that the condition onλ ensures that the weight is an integrable function, and thatλ can be taken to be 1 if p > 1. Thus, the middle term in the foregoing display defines an equivalent norm onBp. We shall also prove the analog of Theorem 1.1 for integral powers of the Szegö kernelSσ.

As an application of Theorem 1.1, we shall obtain a necessary and sufficient condition forhf(µ)andhf(σ)to belong to the Schatten classSpwhen 1≤ p < ∞.

In [Sy1] and [Sy3] it was proved that the conditionf ∈ Bpis a sufficient con- dition forhf to belong toSp (1 ≤ p < ∞) for finite-type domains in C2, for strictly pseudo-convex domains inCn, and also for ellipsoids in Cn. The neces- sity of the condition was left open. Here we show that the condition is necessary when we restrict to the class of ellipsoids that we have defined, and also for strictly pseudo-convex domains.

Using the method of [FR], it was also mentioned in [Sy1] that, forp > 1, hf(µ)Sp implies

Z

q

Sµ−1(z, z) Z

∂q

Sµ2(z, ζ)f(ζ) dµ(ζ)

pB(z, z) dV(z) < ∞. (4) In the case of the unit ball, condition (4) is immediately seen to be equivalent to the fact thatg is in the space Bp. The reason for this is the link between Sσ2and a derivative ofSσ given by (2).

Our result in the case of the ellipsoids is as follows, wherehf stands for both hf(µ)andhf(σ).

Theorem 1.2. Let 1≤ p < ∞. Then hf∈ Spis equivalent tof ∈ Bp.

In the case ofhf(µ), Theorem 1.2 follows directly from Theorem 1.1 and [Sy1] for p > 1. We give a new proof, which extends to the case p = 1. We also prove that Theorem 1.2 is still valid for the Hankel operators based on weighted Bergman projections.

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We next turn to the case of a strictly pseudo-convex domain. We prove the analogs of Theorems 1.1 and 1.2 in this case. In this context the identity (2), which we wrote for the ball, holds in an approximate way. This is an easy consequence of Fefferman’s expansion for the Szegö kernel. Although these results on strictly pseudo-convex domains all follow from somewhat standard techniques, it seems that they never appeared in print before. The idea of approximate identities is also used to deduce the two main theorems in the case of the surface measuredσ on the ellipsoids.

We mention that the charaterization of bounded and compact Hankel operator is known in the case of a strictly pseudo-convex domain. In the case of the unit ball [CRW] and in general [KL] it has been shown thathfis bounded if and only iff ∈ BMO and is compact if and only iff ∈ H2∩ VMO. We also mention that charac- terizations of symbols of big Hankel operators have been obtained in [KLR1] and [BeLi], and that related results appear in [KLR2].

The paper is organized as follows. Sections 2, 3, and 4 are devoted to the prob- lem on ellipsoids with measuredµ. In Section 2 we study the powers of the Szegö kernel, and in Section 3 we prove Theorem 1.1 on equivalence of norms inBp(q).

We believe that these results, and the techniques involved, may have applications beyond what is offered here. In Section 4 we prove Theorem 1.2. In Section 5, we prove the corresponding results for weighted Bergman spaces onq. In Section 6 we consider the case of a smoothly bounded strictly pseudo-convex domain, and in Section 7 we conclude by indicating how to translate the results proved on the ellipsoids to the case of the surface measure.

Finally, we mention that in an upcoming paper [BPS] we study the question of factorization of Hardy spaces as well as characterization of bounded and com- pact Hankel operators on a class of finite type domains inCn that includes the ellipsoids.

2. Powers of the Szegö Kernel

The next three sections deal with the Szegö kernel and the Hankel operator related to the measuredµ, so we shall omit all indices or exponents and write simply S (resp.hf) instead of Sµ(resp.hf(µ)). We shall also write  instead of q. With respect to the surface measuredµ on the boundary ∂ (see (3)), the Szegö kernel S(z, ζ) has expression

S(z, ζ) = cX

m

0¡

m1+mq2 + 1 +q1 0(m1+ 1)0¡m2

q +1qzm¯ζm

= c(1 − z1¯ζ1)−(1+1/q)



1− z2¯ζ2

(1 − z1¯ζ1)1/q

−2 .

(see [BoLo]). We recall that the Bergman kernel also has an explicit expression of this type, which allows us to considerdµ as a natural measure on ∂.

We want to study the integral operatorMλ(λ > 0), given by f 7→

Z

∂Sλ+1(·, ζ)f(ζ) dµ(ζ),

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when acting on holomorphic functions. We can write S(z, ζ) =X

m

γm−1zm¯ζm, withγm=R

∂m|2dµ(ζ). We are going to write Sλ+1(z, ζ) in the same way.

Lemma 2.1. Forl > 0 we have Sl(z, ζ) = cl

X

m

γm−1A(l)mzm¯ζm,

where the sum is taken overm ∈ Z2, m1, m2 ≥ 0, and A(l)m =0¡

m1+mq2 +¡ 1+q1

l 0¡

m1+mq2 + 1 +1q ·0(m2+ 2l)

0(m2+ 1) · 0¡m2

q +q1 0¡m2

q +¡ 1+1q

l.

Proof. We begin by settingSl(z, ζ) =P

mcmzm¯ζm, and we wish to compute the coefficientscm. Notice that

Z

∂Sl(z, ζ)ζmdµ(ζ) = cmzm Z

∂m|2dµ(ζ)

= cmγmzm. Recalling that

Sl(z, ζ) = cl(1 − z1¯ζ1)−l(1+1/q)



1− z2¯ζ2

(1 − z1¯ζ1)1/q

−2l

and using Lemma 1.6 in [BoLo], it is easy to see that A(l)m = cmγm

= cl0 Z

∂(1 − w1)−l(1+1/q)



1− w2

(1 − w1)1/q

−2l

¯w1m1¯wm22dµ(w)

= cl0 0(m2+ 2l)

0(m2+ 1)0(2l)· 0¡

m1+mq2 +¡ 1+1q

l 0¡m2

q +¡ 1+1q

l

0(m1+ 1)

×0(m1+ 1)0¡m2

q +1q 0¡

m1+mq2 + 1 +1q . This proves the lemma.

Thus we have shown that, iff(z) =P

mamzmis inH2(), then Z

∂Sλ+1(z, ζ)f(ζ) dµ(ζ) = cλ

X

m

amA(λ+1)m zm. (5)

We now define the operators that are our main technical tools.

Definition 2.2. Forλ ≥ 0 we define the operators Mλand ˜Mλacting on holo- morphic functions as follows. Letf(z) =P

mamzminH2(). Then we set

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Mλf(z) :=

Z

∂Sλ+1(z, ζ)f(ζ) dµ(ζ) = cλX

m

amA(λ+1)m zm (6) and, with some abuse of notation,

M˜λf(z) :=X

m



m1+m2

q + 1

(1+1/q)λm2

q + 1

(1−1/q)λ

amzm. (7)

Notice that, indeed, ˜MλB ˜Mλ0 = ˜Mλ+λ0.

We have the following result relating the operatorsMλand ˜Mλ.

Lemma 2.3. Letλ ≥ 0 and let dν be a finite measure on  that is invariant under the action of T2on given by

(w1, w2) 7→ (e1w1, e2w2).

Then, for allp > 0 there exists cp > 0 such that, for all holomorphic functions f ∈ H2(),

1 cp

Z

|Mλf |pdν ≤ Z

| ˜Mλf |pdν ≤ cp Z

|Mλf |pdν.

Proof. We setl := λ + 1 and define

αm:= α(l)m := A(l)m

¡m1+mq2 + 1(1+1/q)(l−1)¡m2

q + 1(1−1/q)(l−1). (8) From the invariance ofdν, it follows that

Z

|Mλf |p

= 1

(2π)2 Z 2π

0

Z 2π 0

Z

|Mλf(e1w1, e2w2)|pdν(w1, w2) dθ11; the same holds with ˜Mλ in place ofMλ. If we integrate first in θ1, θ2, we see that the inequalities will follow from the fact that the operator involved on double Taylor series is bounded onHp(T2). It therefore suffices to show that the two se- quencesm)m∈N2 and−1m)m∈N2 define two bounded Fourier multipliers of the spacesHp(T2). We shall prove it for the first sequence (the proof for the second is identical).

If we writeαmas a product then we are led to consider sequences of the type β(1)m = 0(m2+ 1 + s)

0(m2+ 1)(m2+ q)s, β(2)m = 0¡m2

q +1qm2

q +q1s 0¡m2

q +q1+ s , β(3)m = 0¡

m1+mq2 + 1 +1q + s 0¡

m1+mq2 + 1 +1q

m1+mq2 + 1s,

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where in each cases is a positive number related to l and q. It suffices to show that each of these three sequencesm(j)) (j = 1, 2, 3) gives rise to a bounded Fourier multiplier. Let us first look atm(j)) for j = 1, 2. As these sequences depend only onm2, we may restrict to problems of multipliers on the torus of dimension 1.

We now recall the sufficient condition of Mihlin type, which ensures that the sequencen)n∈Nis a Fourier multiplier ofHp(T). We define the difference op- erators1kby induction, setting10 = Id and 1βn= βn− βn+1. Then the Mihlin condition may be written as

|1kβn| ≤ Cl(n + 1)−k, k = 0,1, 2, . . .. (9) If the sequencen)n∈N satisfies (9), then it is a bounded Fourier multiplier of Hp(T) for all p > 0 (see [St, pp. 115, 245]). It is easy to prove that the sequence onN which gives rise to β(1)m satisfies (9), so that it defines a bounded multiplier.

In order to analyze the sequenceβ(2)m , we shall use the following elementary lemma.

Lemma 2.4. Let(βn)n∈Nbe a sequence such that, for eachk = 0,1, . . ., q − 1, the sequence qn+k)n∈N is a Fourier multiplier of Hp(T). Then(βn)n∈N is a Fourier multiplier ofHp(T).

Proof. Letωj (j = 0,1, . . ., q − 1) denote the qth roots of unity, and for f ∈ Hp(T) define

fk(z) := 1 q

Xq−1 j=0

¯ωjkf(ωjz).

Then eachfkis inHp(T) with norm bounded by the norm of f, and f =Pq−1

k=0fk. Moreover,fkmay be written aszkgk(zq), where gkhas Fourier coefficients given by

ˆgk(n) = ˆf(nq + k).

LetT be the operator given by the multiplier (βn)n∈N. Then Tfkis given by the action of the multiplierqn+k)n∈Nongk. It follows from the hypothesis that Tfk

belongs toHp(T). Hence, Tf ∈ Hp(T).

We may now return to the sequence(2)m ). Using Lemma 2.4, it suffices to show that, for eachk = 0,1, . . ., q − 1, the sequence

η(k)n = 0¡

n + k+1q

n + 1 + kqs 0¡

n +k+1q + s

satisfies (9). Notice that (by the Stirling formula) we obtain the asymptotics

n→∞lim 0¡

n + k+1q

n + 1 + kqs 0¡

n +k+1q + s = 1, and(k)n may be written asη(k)n · γn(k), with

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γn(k)= 1 − n +k+1q n + k+1q + s·

¡n + 2 +kqs

¡n + 1 +kqs.

Using these two facts, it is elementary to prove that

|1jγn(k)| ≤ Cj(n + 1)−j, which we wanted to prove to conclude form(2)).

It remains to consider the sequence(3)m ). Using the same kind of argument as in Lemma 2.4, we are lead to consider separately the sequences

δm(k)= 0¡

m1+ m2+ 1 + k+1q + s

¡m1+ m2+ 1 +kqs 0¡

m1+ m2+ 1 +k+1q 

fork = 0,1, . . ., q −1. Moreover, using the same arguments as before, it is easy to see thatδ(k)m = ηm1+m2, where (ηn)n∈Nis a Fourier multiplier of the spacesHp(T).

In order to conclude the proof, we use the following elementary lemma.

Lemma 2.5. Assume that the sequence n)n∈N is a Fourier multiplier of the spaceHp(T). Then (δm)m∈N2, defined by δm= ηm1+m2, is a Fourier multiplier of Hp(T2).

Proof. LetF(θ1, θ2) =P

m1≥0,m2≥0am1,m2ei(m1θ1+m2θ2)be a polynomial. Then F(θ1, θ2) = G(θ1, θ2− θ1)

with

G(θ1, θ2) = X

n1≥0

X

0≤n2≤n1

bn1,n2ei(n1θ1+n2θ2)

if we choosebn1,n2 = an1−n2,n2. Multiplication by δm foram1,m2 becomes multi- plication byηn1 forbn1,n2. Therefore, the muliplier acts on the first variable for G, and the lemma follows from the fact that F and G have the same norm in Hp(T2).

This ends the proof of Lemma 2.3.

Roughly speaking, Lemma 2.3 shows that the operatorMλacts on holomorphic functions like ˜Mλ. In order to understand the action of ˜Mλ, recall that N is the differential operatorN := z1z1+zq2z2. For a holomorphic function

f(z) =X

m

amzm, we have the equalities

(I + N )f(z) =X

m



m1+m2

q + 1

 amzm,

 I + z2

q z2



f(z) =X

m

m2

q + 1

 amzm.

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Hence, the action of ˜Mλon holomorphic functions may be seen as the product of fractional powers ofI + N and I + zq2z2. As it is easier to deal with differen- tial operators than fractional powers, we shall make use of ˜Mqk(k ∈ N, k ≥ 1). A simple calculation now shows that

M˜qkf(z) =X

m



m1+m2

q + 1

(1+q)k 1+m2

q

(q−1)k amzm

=

 I +z2

q z2

(q−1)k

(I + N )(q+1)kf(z), which is a classical differential operator.

We conclude this section by recalling some geometrical facts about our domains and kernels.

As is well known, on the boundary∂ of  there exists a non-isotropic pseudo- metricdb(as defined in [NRSW, Def. (1.1)]) for all finite type domains ofC2. In our case, one may use a simple expression fordb(see [BoLo]). Thus, we set

db(z, w) = =¡ ¯w1(w1− z1) + q ¯w2|w2|2q−2(w2− z2)

+ |w2|2q−2|w − z|2+ |w − z|2q. (10) Furthermore, there exists a tubular neighborhood of the boundary∂ such that eachz ∈ U has a unique normal projection π(z) on ∂. For z, w ∈ U we set

d(z, w) := δ(z) + δ(w) + db(π(z), π(w)), (11) where, we recall,δ(z) := 1 − |z1|2− |z2|2q. Moreover, we set

τ(w, r) := min{r1/2|w2|1−q, r1/2q}. (12) For the estimates for the Szegö and the Bergman kernels on the diagonal, one has:

S(z, z) ' (1 − |z1|2)−(1+1/q)



1− |z2|2 (1 − |z1|2)1/q

−2

, (13)

B(z, z) ' (1 − |z1|2)−(2+1/q)



1− |z2|2 (1 − |z1|2)1/q

−3

. (14)

These formulas may be found in [BoLo]. They have also been proved in the con- text of pseudo-convex domains of finite type by Catlin in [Ca]. In order to recover the well-known Catlin’s estimates—that is,

B(z, z) ' δ(z)−2· τ(z)−2, S(z, z) ' δ(z)−1· τ(z)−2,

where, for convenience of notation, we writeτ(z) := τ(z, δ(z)) (see (12))—it suf- fices to check that on these ellipsoids one has

τ(z) '¡

(1 − |z1|2)1/q− |z2|21/2

and δ(z) ' (1 − |z1|2)1−1/qτ(z)2. (15)

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3. Equivalence of Norms

In this section we prove the equivalence of norms in the Besov space—that is, The- orem 1.1. We first prove the inequality on the right in Theorem 1.1. We do this by proving a slightly more general fact. We should emphasize the fact that this part of Theorem 1.1 is valid in the larger context of domains of finite type inC2 (and of convex domains of finite type inCn) because it uses only size estimates on the Szegö kernel. The results of Section 2 will only be used in the reverse inequality.

We recall the notation introduced in (6):

Mλf(z) = Z

∂Sλ+1(z, ζ)f(ζ) dµ(ζ).

On the ellipsoid we define the differential operator N := z1z1+z2

q z2, (16)

and forβ ∈ R we denote by βthe numberβ/2q for β ≥ 0 and β/2 for β < 0.

Recall that

τβ≤ cδβ.

Proposition 3.1. Letl be a nonnegative integer, α, β ∈ R, λ ≥ 0, and p > 0.

We assume that the inequalities

α + β+ lp + 1 > 0, λp + α + 1 + (2λp + β) > 0

are satisfied. Then there exists a constantc > 0 such that, for all holomorphic functionsf,

Z

|Mλf(z)|pS−pλ(z, z)δα(z)τβ(z) dV(z)

≤ c Z



X

0≤k≤l

|∇kf(z)|pδlp+α(z)τβ(z) dV(z).

Notice that the conditions onα, β, l, λ, p are equivalent to the integrability of the weight, and that the right-hand side inequality of Theorem 1.1 corresponds to the caseα = β = −2.

Proof. The proof of Proposition 3.1 is given in three steps. We first give a new expression forMλf(z), which is obtained by integrations by parts. Then we con- sider the casesp > 1 and p ≤ 1 separately. The new expression is based on the following lemma.

Lemma 3.2. Letf, g be holomorphic functions in H2(). Then if we set Dn= N +¡

1+n+1q 

I for all nonnegative integer n, the following identities hold:

Z

∂f(u)g(u) dµ(u) = c00

Z

f(u)D0g(u) dV(u), (17) Z

f(u)g(u)δn(u) dV(u) = cn0 Z

f(u)Dng(u)δn+1(u) dV(u). (18)

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Proof. We only prove the identity (17), the proof for (18) being analogous. This identity, once given the Taylor series off and g, is an easy consequence of (3).

Indeed, recall that

kumk2L2(dµ)= c0

0(m1+ 1)0¡m2+1

q

 0¡

m1+ 1 +m2q+1 , (see [BoLo]). Writef(u) =P

mamumandg(u) =P

mbmum. Then Z

∂f(u)g(u) dµ(u) = c0

X

m

am¯bmkumk2L2(dµ). Moreover,kumk2L2(dV )= c10(m1+ 1)0¡m2+1

q

0¡

m1+ 1 +m2q+1+ 1−1 , and Z

f(u)D0g(u) dV(u) = c000

X

m

am¯bm



m1+ 1 +m2+ 1 q



kumk2L2(dV ). Recalling that0(z + 1) = z0(z), we obtain (17).

If we use Lemma 3.2k + l times, starting from the definition of Mλf(z), we ob- tain the following.

Lemma 3.3. Letk, l ∈ N. Then there exists Ck,l such that, forf ∈ H2(), the following identity holds:

Mλf(z)

= Ck,l Z



D¯k,ζ· · · ¯D0Sλ+1(z, ζ)Dk+l· · · Dk+1f(ζ)δk+l(ζ) dV(ζ). (19)

We now prove Proposition 3.1 for 1< p < ∞. Denote by Tk(λ)the integral oper- ator defined by

Tk(λ)g(z) = δλ(z)τ2λ(z) Z



D¯k,ζ· · · ¯D0Sλ+1(z, ζ)g(ζ)δk(ζ) dV(ζ).

It is sufficient to show that, fork large enough and for some constant c, we have Z

|Tk(λ)g(z)|pδα(z)τβ(z) dV(z) ≤ c Z

|g(z)|pδα(z)τβ(z) dV(z).

for all (not necessarily holomorphic) functionsg. In order to prove such an esti- mate, it suffices to show that the integral kernel of the operatorTk(λ)onLpατβdV ) satisfies the assumptions of Schur’s lemma. Notice that the kernelKk(z, ζ) has expression

Kk(z, ζ) = δλ(z)τ2λ(z) ¯Dk,ζ· · · ¯D0Sλ+1(z, ζ)δk−α(ζ)τ−β(ζ). (20) From the estimates in [NRSW], we know that

| ¯Dk,ζ· · · ¯D0Sλ+1(z, ζ)| ≤ cd(z, ζ)−2−λ−kτ(z, d(z, ζ))−2−2λ, (21) whered(z, u) is defined as in (11). We can apply Schur’s lemma with the function δaτbby using the following lemma, which relies on the estimates (21).

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Lemma 3.4. Under the condition

λp + α + 1 + (2λp + β)> 0, fork large enough one can find a, b such that

Z

|Kk(z, ζ)|δap0(ζ)τbp0(ζ) dV(ζ) ≤ cδap0(z)τbp0(z), Z

|Kk(z, ζ)|δap+α(z)τbp+β(z) dV(z) ≤ cδap(ζ)τbp(ζ).

Proof. The lemma is an elementary consequence of the following inequality, whose proof may be found in [Sy2]:

Z

d(z, ζ)aτ(z, d(z, ζ))bδα(ζ)τβ(ζ) dV(ζ) ≤ cδα+a+2(z)τβ+b+2(z) (22) under the conditions

α + β+ 1 > 0 and − α − a − 2 + (−β − b − 2) > 0.

We recall that from (20) and (21) we have the estimate

|Kk(z, ζ)| ≤ cδλ(z)τ2λ(z)d(z, ζ)−2−λ−kτ(z, d(z, ζ))−2−2λδk−α(ζ)τ−β(ζ).

Thus, according to (22), we must show that there exists(a, b) such that

−ap0+ λ + (−bp0+ 2λ)> 0, ap + α + λ + (bp + β + 2λ)+ 1 > 0.

If these two conditions are satisfied then the previous two are also satisfied fork large enough.

Now(a, b) satisfies the first condition if it is in a convex cone whose vertex is the point(λ/p0, 2λ/p0). Analogously, the second condition is satisfied if (a, b) is inside a convex cone having vertex in(a0, b0), where

a0= −α + λ + 1

p , b0 = −β + 2λ

p .

In order for these two regions to intersect, it suffices that the vertex of first cone belongs to the second one:

α

p0p + α + λ +

2λ

p0p + β + 2λ



+ 1 > 0;

that is,

λp + α + (2λp + β)+ 1 > 0, which is what we wished to show.

This finishes the proof of Proposition 3.1 in the case 1< p < ∞.

Now let 0 < p ≤ 1. Let (w(j)) denote a sequence of points such that poly- discs of typeQj = Q(w(j), ηδ(w(j)) give a Whitney covering of  as well as

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2Qj := Q(w(j), 2ηδ(w(j))). If we write g = Dk+l· · · Dk+1f(ζ) then we see that

|Mλf(z)| is equivalent to the sum over j of

d(z, w(j))−2−λ−kτ(z, d(z, w(j)))−2−2λδk+l(w(j)) Z

Qj

|g(ζ)| dV(ζ).

We have used the fact that all functionsd(z, ·), δ, τ are essentially constant in- sideQj. It follows from the subharmonicity of g (and the fact that p ≤ 1) that

|Mλf(z)|pcan be majorized by a constant times X

j

d(z, w(j))(−2−λ−k)pτ(z, d(z, w(j)))(−2−2λ)pδ(k+l)p(w(j))

ס

Vol(Qj)p−1Z

2Qj

|g(ζ)|pdV(ζ).

Now, fork large enough, we can apply inequality (22) to obtain the bound Z

d(z, w(j))(−2−λ−k)pτ(z, d(z, w(j)))(−2−2λ)pδλp+α(z)τ2λp+β(z) dV(z)

≤ cδ−(k+2)p+α+2(w(j)−2p+β+2(w(j)).

From these two last inequalities and using the Whitney property of the covering 2Qj, we see that

Z

|Mλf(z)|pδλp+α(z)τ2λp+β(z) dV(z)

≤ cX

j

δlp+α(w(j)β(w(j)) Z

2Qj

|g(ζ)|pdV(ζ)

≤ c Z

|δ(z)g(z)|pδα(z)τβ(z) dV(z).

This finishes the proof of Proposition 3.1.

Now we make use of our work in Section 2 relating the operatorsMλand ˜Mλ. The proof of the next result is an immediate consequence of Lemma 2.3 and Proposi- tion 3.1.

Corollary 3.5. Letl be a nonnegative integer, α, β ∈ R, λ ≥ 0, and p > 0.

Suppose that the inequalities

α + β+ lp + 1 > 0 and λp + α + 1 + (2λp + β) > 0

are satisfied. Then there exists a constantc > 0 such that, for all holomorphic functionsf,

Z

| ˜Mλf(z)|pS−pλ(z, z)δα(z)τβ(z) dV(z)

≤ c Z



X

0≤k≤l

|∇kf(z)|pδlp+α(z)τβ(z) dV(z).

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We now prove the left inequality in Theorem 1.1. Assume that Z

q

|Mλg(z)|pS−λp(z, z)B(z, z) dV(z) < ∞.

Then the same is valid with ˜Mλinstead ofMλ. We recall that M˜λB ˜Mλ0 = ˜Mλ+λ0.

Using Corollary 3.5 withλ replaced by kq − λ, with f replaced by ˜Mλg, and with l = 0 and α, β suitably chosen, for k large enough we find that

Z

q

| ˜Mkqg(z)|pδkqp−2(z)τ2kqp−2dV(z) < ∞.

Remember that ˜Mkq

I +zq2z2

(q−1)k

(I +N )(q+1)k. In order to be able to prove thatg is in the Besov space Bp with control of the norm, it suffices to prove the following lemma.

Lemma 3.6. Letα, β ∈ R such that α + β+ 1 > 0. Given any p > 0, there exists a constantc > 0 such that, for all holomorphic functions g,

Z

|g(z)|pδα(z)τβ(z) dV(z) ≤ c Z

|(I + z2z2)kg(z)|pδα(z)τβ+2kp(z) dV(z).

Proof. It suffices to prove the lemma fork = 1. Using the estimate (15), we have Z

|g(z)|pδα(z)τβ(z) dV(z)

≤ c Z

|z1|<1(1 − |z1|2)α(1−1/q)

× Z

|z2|2<(1−|z1|2)1/q

¡(1 − |z1|2)1/q− |z2|2α+β/2

|g(z)|pdV(z2) dV(z1).

Then, by applying Hardy’s lemma in the inner integral (see e.g. [AFJP, Thm. 6]), we see that the integral

Z

|z2|2<(1−|z1|2)1/q

¡(1 − |z1|2)1/q− |z2|2α+β/2

|g(z)|pdV(z2) is bounded by a constant times

Z

|z2|2<(1−|z1|2)1/q

¡(1 − |z1|2)1/q− |z2|2α+β/2+p

|(I + z2z2)g(z)|pdV(z2).

Therefore, Z

|g(z)|pδα(z)τβ(z) dV(z) ≤ c Z

|(I + z2z2)g(z)|pδα(z)τβ+2p(z) dV(z), as follows again from the estimate (15).

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4. Necessary Conditions for Schatten Class Hankel Operators

We now prove Theorem 1.2 for the operatorhf = hf(µ). For 1 < p < ∞ it fol- lows immediately from results in [Sy1, Sec. 4]. We now give another proof, which holds also forp = 1. We shall prove that

khfkpSp ≥ c Z



S−3(z, z) Z

∂S4(z, ζ)f(ζ) dµ(ζ)

pB(z, z) dV(z).

Recall that, for 1≤ p < +∞, any bounded operator T, and any orthonormal sequence(aj), we have that

X

j

|hTaj, aji|p≤ kT kpSp. (23) Moreover, it holds that

kXT Y kSp ≤ kXkkT kSpkY k, (24) for any bounded operatorsX, Y.

As before, let(w(j)) be a sequence of points such that Qj := Q(w(j), ηδ(w(j))) is a Whitney covering of and ˜Qj = Q(w(j), ηδ(w(j))/C0) are pairwise dis- joint. The size ofQj in the complex transverse directionNw(j) isηδ(w(j)), and it isητ(w(j)) in the complex tangential direction (see [Sy1]). Let Bj = π(Qj), whereπ denotes the normal projection of a tubular neighborhood of the bound- ary onto the boundary itself. InsideQj, the quantity S−3(z, z)B(z, z) is of the orderµ(Bj)3· Vol(Qj)−1. Moreover, using the mean value property, from stan- dard techniques it follows that—forη small enough and α, β fixed in such a way thatα + β+ 1 > 0—for F holomorphic one has the equivalence

Z

|F(z)|pδα(z)τβ(z) dV(z) 'X

j

|F(w(j))|pδα(w(j)β(w(j)) Vol(Qj) (25)

(see [CRW] in the case of the unit disk and [Sy2] for its generalization in our con- text). In particular, we have

Z



S−3(z, z) Z

∂S4(z, ζ)f(ζ) dµ(ζ)

pB(z, z) dV(z)

≤ cX

j

µ(Bj)3p Z

∂S4(w(j), ζ)f(ζ) dµ(ζ)

p. (26) If we setej := µ(Bj)3/2S2(·, w(j)), the right-hand side of (26) is equal to

cX

j

Z

∂hf(ej)ej

p = cX

j

|hhf(ej), eji|p.

Now we claim that the sequence(ej) is the image under a bounded operator Y : L2(dV ) → L2(∂) of an orthogonal system (aj) in L2(dV ), with kajkL2(dV )' 1. Assuming the claim for the moment, we finish the proof.

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