Skew Carleson measures in strongly pseudoconvex domains

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Skew Carleson measures in strongly pseudoconvex domains

Marco Abate, Jasmin Raissy

To cite this version:

Marco Abate, Jasmin Raissy. Skew Carleson measures in strongly pseudoconvex domains . Complex

Analysis and Operator Theory, Springer Verlag, 2018, �10.1007/s11785-018-0823-4�. �hal-01610760�

arXiv:1710.01631v1 [math.CV] 4 Oct 2017

DOMAINS

MARCO ABATE AND JASMIN RAISSY*

Abstract. Given a bounded strongly pseudoconvex domainDinCⁿwith smooth boundary, we give a characterization through products of functions in weighted Bergman spaces of (λ, γ)-skew Carleson measures onD, withλ >0 andγ >1−_n+1¹ .

1. Introduction

Carleson measures are a powerful tool and an interesting object to study. They have been introduced by Carleson [6] in his celebrated solution of the corona problem to study the structure of the Hardy spaces of the unit disc ∆⊂C. LetAbe a Banach space of holomorphic functions on a domainD⊂Cⁿ; givenp≥1, a finite positive Borel measureµonDis aCarleson measure ofA (for p) if there is a continuous inclusion A ֒→L^p(µ), that is there exists a constantC > 0 such that

∀f ∈A Z

D

|f|^pdµ≤Ckfk^p_A.

In this paper, we are interested in Carleson measures for Bergman spaces, that is spaces ofL^p holomorphic functions, usually denoted byA^p(relationships between Carleson measures for Hardy spaces and Carleson measures for Bergman spaces can be found in [5]). Carleson measures for Bergman spaces have been studied by several authors, including Hastings [11] (see also Oleinik and Pavlov [22] and Oleinik [21]) for the Bergman spacesA^p(∆), Cima and Wogen [8] in the case of the unit ballBⁿ⊂Cⁿ, Zhu [25] in the case of bounded symmetric domains, Cima and Mercer [7]

for Bergman spaces in strongly pseudoconvex domainsA^p(D), and Luecking [19] for more general domains.

Given D⊂⊂Cⁿ a bounded strongly pseudoconvex domain in Cⁿ with smooth C^∞ boundary, a positive finite Borel measureµonD and 0< p <+∞, we denote by L^p(µ) the set of complex- valuedµ-measurable functionsf:D→Csuch that

kfkp,µ:=

Z

D

|f(z)|^pdµ(z) ^1/p

<+∞.

Ifµ=δ^αν for someα∈R, whereδ(z) =d(z, ∂D) is the distance from the boundary ofD andν is the Lebesgue measure, theweighted Bergman space is defined as

A^p(D, α) =L^p(δ^αν)∩ O(D),

2010 Mathematics Subject Classification: 32A36 (primary), 32A25, 32Q45, 32T15, 46E22, 46E15, 47B35 (sec- ondary).

Keywords: Carleson measure; Toeplitz operator; strongly pseudoconvex domain; weighted Bergman space.

∗Partially supported by ANR project LAMBDA, ANR-13-BS01-0002 and by the FIRB2012 grant “Differential Geometry and Geometric Function Theory”, RBFR12W1AQ 002.

1

where O(D) denotes the space of holomorphic functions on D, endowed with the topology of uniform convergence on compact subsets. Together with Saracco, we gave in [3] a characterization of Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain.

It is a natural question to study Carleson measures for different exponents, that is the embedding of weighted Bergman spaces A^p(D, α) into L^q spaces. Given, 0 < p, q < +∞ and α > −1, a finite positive Borel measure µ is called a (p, q;α)-skew Carleson measure if A^p(D, α) ֒→ L^q(µ) continuously, that is there exists a constantC >0 such that

Z

D

|f(z)|^qdµ(z)≤Ckfk^q_p,α

for allf ∈A^p(D, α). Investigation on (p, q;α)-skew Carleson measure has been started by Luecking in [20] and recently extended by Hu, Lv and Zhu in [14], where these measures are called (p, q, α) Bergman Carleson measures. It turns out (see [14] and the next section for details) that the property of being (p, q;α)-skew Carleson depends only on the quotientq/p and onα, allowing us to define (λ, γ)-skew Carleson measures forλ >0 andγ >1−_n+1¹ . Roughly speaking, a measure is (λ, γ)-skew Carleson if and only if it is a (p, q; (n+ 1)(γ−1))-skew Carleson measure for some (and hence any)p, q such thatq/p=λ(see Definition 2.17).

The main result of this paper gives a characterization of (λ, γ)-skew Carleson measures on bounded strongly pseudoconvex domains through products of functions in weighted Bergman spaces.

Theorem 1.1. Let D⊂⊂Cⁿ be a bounded strongly pseudoconvex domain, and let µbe a positive finite Borel measure on D. Fix an integer k ≥1, and let 0 < pj, qj <∞ and 1−_n+1¹ < θj be given, forj= 1, . . . , k. Set

λ= Xk

j=1

qj

pj

and γ= 1 λ

Xk

j=1

θj

qj

pj

.

Then µis a(λ, γ)-skew Carleson measure if and only if there existsC >0 such that (1.1)

Z

D

Yk

j=1

|fj(z)|^qjdµ(z)≤C Yk

j=1

kfjk^q_p^j

j,(n+1)(θj−1)

for any fj∈A^pj D,(n+ 1)(θj−1) .

This result generalizes the analogue one obtained by Pau and Zhao in [23] on the unit ball ofCⁿ. The proof relies on the properties of two closely related operators. The first one is a Toeplitz-like operatorT_µ^β (see (3.1)), depending on a parameterβ ∈N^∗ and on a finite positive Borel measure µ, and the main issue consists in identifying functional spaces that can act as domain and/or codomain of such an operator. The second operator, S_t,µ^s,r (see (3.2)), depends on µ and three positive real parametersr, s,t >0, and its norm can be used to bound the norm of the operators T_µ^β, under suitable assumptions. In particular, the key step in the proof of the necessity implication in the case 0< λ <1 consists in finding criteria for a measure to be (λ, γ)-skew Carleson. These criteria are expressed in terms of mapping properties of the two operators T_µ^β and S^s,r_t,µ in the technical Propositions 3.4 and 3.6.

The paper is structured as follows. In Section 2 we shall collect the preliminary results and definitions. In Section 3 we shall study the properties of the operatorsT_µ^β andS_t,µ^s,r and prove our main result.

2. Preliminary results

In this section we collect the precise definitions and preliminary results we shall need in the rest of the paper.

From now on,D ⊂⊂Cⁿ will be a bounded strongly pseudoconvex domain inCⁿ with smooth C^∞boundary. Furthermore, we shall use the following notations:

• δ:D → R⁺ will denote the Euclidean distance from the boundary of D, that is δ(z) = d(z, ∂D);

• given two non-negative functions f,g:D→R⁺ we shall writef g to say that there is C >0 such thatf(z)≤Cg(z) for allz∈D. The constantCis independent ofz∈D, but it might depend on other parameters (r,θ, etc.);

• given two strictly positive functionsf,g:D→R⁺we shall writef ≈giff gandgf, that is if there isC >0 such thatC⁻¹g(z)≤f(z)≤Cg(z) for all z∈D;

• ν will be the Lebesgue measure;

• O(D) will denote the space of holomorphic functions onD, endowed with the topology of uniform convergence on compact subsets;

• given 0< p <+∞, theBergman spaceA^p(D) is the (Banach ifp≥1) spaceL^p(D)∩O(D), endowed with theL^p-norm;

• more generally, if µ is a positive finite Borel measure on D and 0 < p < +∞ we shall denote byL^p(µ) the set of complex-valuedµ-measurable functionsf:D→Csuch that

kfkp,µ:=

Z

D

|f(z)|^pdµ(z) ^1/p

<+∞.

Ifµ=δ^βν for some β∈R, we shall denote byA^p(D, β) theweighted Bergman space A^p(D, β) =L^p(δ^βν)∩ O(D),

and we shall writek · kp,β instead ofk · k_p,δ^β_ν;

• K:D×D→Cwill be the Bergman kernel of D;

• for eachz0∈D we shall denote bykz0:D→Cthenormalized Bergman kernel defined by kz0(z) = K(z, z0)

pK(z0, z0) = K(z, z0) kK(·, z0)k2 ;

• givenr∈(0,1) andz0 ∈D, we shall denote byBD(z0, r) the Kobayashi ball of centerz0

and radius ¹₂log^1+r_1−r.

We refer to, e.g., [1, 2, 15, 16], for definitions, basic properties and applications to geometric function theory of the Kobayashi distance; and to [13, 12, 17, 24] for definitions and basic properties of the Bergman kernel.

Let us now recall a number of results we shall need on the Kobayashi geometry of strongly pseudoconvex domains.

Lemma 2.1([18, Corollary 7],[4, Lemma 2.1]). LetD⊂⊂Cⁿbe a bounded strongly pseudoconvex domain, andr∈(0,1). Then

ν BD(·, r)

≈δⁿ⁺¹, (where the constant depends on r).

Lemma 2.2 ([4, Lemma 2.2]). Let D ⊂⊂Cⁿ be a bounded strongly pseudoconvex domain. Then there isC >0 such that

1−r

C δ(z0)≤δ(z)≤ C 1−rδ(z0)

for allr∈(0,1),z0∈D andz∈BD(z0, r).

We shall also need the existence of suitable coverings by Kobayashi balls:

Definition 2.3. LetD ⊂⊂Cⁿ be a bounded domain, andr > 0. Ar-lattice inD is a sequence {ak} ⊂D such that D=S

kBD(ak, r) and there existsm >0 such that any point in D belongs to at mostmballs of the formBD(ak, R), whereR= ¹₂(1 +r).

The existence ofr-lattices in bounded strongly pseudoconvex domains is ensured by the following result:

Lemma 2.4 ([4, Lemma 2.5]). Let D ⊂⊂Cⁿ be a bounded strongly pseudoconvex domain. Then for everyr∈(0,1)there exists anr-lattice inD, that is there existm∈Nand a sequence{ak} ⊂D of points such that D = S∞

k=0BD(ak, r) and no point of D belongs to more than m of the balls BD(ak, R), whereR=¹₂(1 +r).

We shall use a submean estimate for nonnegative plurisubharmonic functions on Kobayashi balls:

Lemma 2.5([4, Corollary 2.8]). LetD⊂⊂Cⁿ be a bounded strongly pseudoconvex domain. Given r∈(0,1), setR=¹₂(1 +r)∈(0,1). Then there exists a constantKr>0 depending onrsuch that

∀z0∈D∀z∈BD(z0, r) χ(z)≤ Kr

ν(BD(z0, r)) Z

BD(z0,R)

χ dν for every nonnegative plurisubharmonic functionχ: D→R⁺.

We shall also need a few estimates on the behavior of the Bergman kernel. The first one is classical (see, e.g., [12]):

Lemma 2.6. Let D⊂⊂Cⁿ be a bounded strongly pseudoconvex domain. Then kK(·, z0)k2=p

K(z0, z0)≈δ(z0)^−(n+1)/2 and kkz0k2≡1 for allz0∈D.

A similar estimate but with constants uniform on Kobayashi balls is the following:

Lemma 2.7 ([18, Theorem 12], [4, Lemma 3.2 and Corollary 3.3]). Let D ⊂⊂Cⁿ be a bounded strongly pseudoconvex domain. Then for everyr∈(0,1)there existcr>0 andδr>0such that if z0∈D satisfiesδ(z0)< δr then

cr

δ(z0)ⁿ⁺¹ ≤ |K(z, z0)| ≤ 1 crδ(z0)ⁿ⁺¹

and cr

δ(z0)ⁿ⁺¹ ≤ |kz0(z)|²≤ 1 crδ(z0)ⁿ⁺¹ for allz∈BD(z0, r).

Remark 2.8. Note that in the previous lemma the estimates from above hold even whenδ(z0)≥δr, possibly with a different constantcr. Indeed, when δ(z0)≥δr andz ∈BD(z0, r) by Lemma 2.2 there is ˜δr>0 such thatδ(z)≥δ˜r; as a consequence we can findMr>0 such that|K(z, z0)| ≤Mr

as soon asδ(z0)≥δrandz∈BD(z0, r), and the assertion follows from the fact thatDis a bounded domain.

A very useful integral estimate is the following:

Proposition 2.9([18, Corollary 11, Theorem 13],[3, Theorem 2.7]). Let D⊂⊂Cⁿ be a bounded strongly pseudoconvex domain, andz0∈D. Let0< p <+∞and−1< β <(n+ 1)(p−1). Then

Z

D

|K(z, w)|^pδ(w)^βdν(w)δ(z)β−(n+1)(p−1)

and Z

D

|kz(w)|^pδ(w)^βdν(w)δ(z)^{β−(n+1)(p2−1)}.

Finally, the normalized Bergman kernel can be used to build functions belonging to suitable weighted Bergman spaces:

Lemma 2.10([14, Lemma 2.6]). Letp >0andθ >1−_n+1¹ be given, and letα= (n+ 1)(θ−1)>

−1. Take β∈N such thatβp >max{θ,(p−1)_n+1ⁿ +θ} and put τ= (n+ 1)

β 2 −θ

p

.

For each a∈D setfa =δ(a)^τk^β_a. Let{ak} be an r-lattice andc={ck} ∈ℓ^p, and put f =

X∞ k=0

ckfa_k. Then f ∈A^p(D, α)with kfkp,α kckp.

We also need to recall a few definitions and results about Carleson measures.

Definition 2.11. Let 0< p,q <+∞andα >−1. A (p, q;α)-skew Carleson measure is a finite positive Borel measureµsuch that

Z

D

|f(z)|^qdµ(z) kfk^q_p,α

for allf ∈A^p(D, α). In other words,µis (p, q;α)-skew Carleson ifA^p(D, α)֒→L^q(µ) continuously.

In this case we shall denote bykµkp,q;α the operator norm of the inclusionA^p(D, α)֒→L^q(µ).

Remark 2.12. When p = q we recover the usual (non-skew) notion of Carleson measure for A^p(D, α).

Definition 2.13. Letθ∈R, and letµbe a finite positive Borel measure onD. Givenr∈(0,1), let ˆµr,θ:D→Rbe defined by

ˆ

µr,θ(z) = µ BD(z, r) ν BD(z, r)θ ; we shall write ˆµr for ˆµr,1.

We say thatµis ageometricθ-Carleson measure if ˆµr,θ∈L^∞(D) for allr∈(0,1), that is if for everyr >0 we have

µ BD(z, r)

ν BD(z, r)θ

for allz∈D, where the constant depends only onr.

Notice that Lemma 2.1 yields

(2.1) µˆr,θ≈δ−(n+1)(θ−1)µˆr.

In [3] we proved (among other things) that, ifp≥1, a measureµis (p, p;α)-skew Carleson if and only if it is geometricθ-Carleson, whereα= (n+ 1)(θ−1). Hu, Lv and Zhu in [14] have given

a similar geometric characterization of (p, q;α)-skew Carleson measures for all values of pandq;

to state their results we need another definition.

Definition 2.14. Letµbe a finite positive Borel measure onD, ands >0. TheBerezin transform oflevel sofµis the functionB^sµ:D→R⁺∪ {+∞}given by

B^sµ(z) = Z

D

|kz(w)|^sdµ(w).

The geometric characterization of (p, q;α)-skew Carleson measures is different according to whetherp≤qorp > q. We first state the characterization for the casep≤q.

Theorem 2.15. Let D⊂⊂Cⁿ be a bounded strongly pseudoconvex domain. Let0< p≤q <+∞

and1−_n+1¹ < θ; setα= (n+ 1)(θ−1)>−1. Then the following assertions are equivalent:

(i) µis a(p, q;α)-skew Carleson measure;

(ii) µis a geometric ^q_pθ-Carleson measure;

(iii) there existsr0∈(0,1)such that µˆr0,^q_pθ∈L^∞(D);

(iv) for everyr∈(0,1)and for every r-lattice {ak} inD we have µ BD(ak, r)

ν BD(ak, r)^q_p^θ

; (v) there existsr0∈(0,1)and ar0-lattice{ak} inD such that

µ BD(ak, r0)

ν BD(ak, r0)^q_pθ

; (vi) for some (and hence all) s > θ^q_p we have

B^sµ(a)δ(a)⁽ⁿ⁺¹⁾(^θqp−^s₂) ;

(vii) there existsC >0 such that for some (and hence all)t >0 we have Z

D

|K(z, a)|^θpq+n+1t dµ(z)δ(a)^−t. Moreover we have

(2.2) kµkp,q;α≈ kµˆ_r,^q

pθk∞≈ kδ^{−(n+1)(pqθ−1)}µˆrk∞≈ kδ⁽ⁿ⁺¹⁾(^s2−θ_p^q)B^sµk∞.

Proof. The equivalence of (i)–(vi), as well as the equivalence for the norms, follows from [14, Theorem 3.1] (and the equivalence of (ii)–(v) was already in [3]).

Now, by Lemma 2.6, (vi) is equivalent to Z

D

|K(z, a)|^sdµ(z)δ(a)⁽ⁿ⁺¹⁾(^θqp−s). Settingt= (n+ 1)

s−θ^q_p

, which is positive if and only ifs > θ^q_p, we see that (vi) is equivalent

to Z

D

|K(z, a)|^θpq+n+1t dµ(z)δ(a)^−t,

that is to (vii).

The geometric characterization of (p, q;α)-skew Carleson measures when p > q has a slightly different flavor:

Theorem 2.16. Let D⊂⊂Cⁿ be a bounded strongly pseudoconvex domain. Let0< q < p <+∞

and1−_n+1¹ < θ; putα= (n+ 1)(θ−1)>−1. Then the following assertions are equivalent:

(i) µis a(p, q;α)-skew Carleson measure;

(ii) ˆµrδ^−αqp ∈L^p−qp (D)for some (and hence any)r∈(0,1);

(iii) ˆµr,θ∈L^p−qp (D, α)for some (and hence any)r∈(0,1);

(iv) ˆµ_r,θ^q

p ∈L^p−qp D,−(n+ 1)

for some (and hence any)r∈(0,1);

(v) for some (and hence any)r∈(0,1) and for some (and hence any)r-lattice {ak} in D we have{µˆr,θ_p^q(ak)} ∈ℓ^p−qp ;

(vi) for some (and hence any)r∈(0,1) and for some (and hence any)r-lattice {ak} in D we have{µˆr(ak)δ(ak)⁽ⁿ⁺¹⁾(^1−θqp)} ∈ℓ^p−qp ;

(vii) for some (and hence all) s > θ^q_p+_n+1ⁿ 1−^q_p

we have δ⁻⁽ⁿ⁺¹⁾(^θqp−^s₂)B^sµ∈L^p−qp D,−(n+ 1)

; (viii) for some (and hence all) s > θ^q_p+_n+1ⁿ

1−^q_p

we have δ⁻⁽ⁿ⁺¹⁾(^θ−s2)B^sµ∈L^p−qp (D, α) ; (ix) for some (and hence all) s > θ^q_p+_n+1ⁿ

1−^q_p

we have δ⁻⁽ⁿ⁺¹⁾(^θqp−^s₂+^p−q_p )B^sµ∈L^p−pq(D) ; (x) for some (and hence all) t >(n+ 1)

1−^q_{p n+1}ⁿ −θ

we have

δ^t Z

D

|K(·, w)|^θ+n+1t dµ(w)∈L^p−qp (D, α). Moreover we have

(2.3) kµkp,q;α≈ kδ^{−(n+1)(θ−s2)}B^sµk ^p

p−q,α≈ kδ−(n+1)(θ−1)^q_pµˆrk ^p

p−q

Proof. The equivalence of (i), (ii), (vi) and (ix), as well as the equivalence of the norms, is in [14, Theorem 3.3].

Recalling that, by Lemma 2.1, ˆµr,θ≈µˆrδ^{(n+1)(1−θ)}, it is easy to see that the equalities

−(n+ 1)(θ−1)q p

p

p−q = (n+ 1)(1−θ) p

p−q+ (n+ 1)(θ−1)

= (n+ 1)

1−θq p

p

p−q−(n+ 1) yield the equivalence of (ii), (iii) and (iv).

The fact that ˆµr,θ≈µˆrδ^{(n+1)(1−θ)}immediately yields the equivalence between (v) and (vi).

The equalities

−(n+ 1)

θq p−s

2 p

p−q−(n+ 1) =−(n+ 1) θ−s

2 p

p−q + (n+ 1)(θ−1)

=−(n+ 1)

θq p−s

2 +p−q p

p p−q yield the equivalence of (vii), (viii) and (ix).

Finally, by Lemma 2.6, (viii) is equivalent to δ−(n+1)(θ−s)Z

D

|K(·, w)|^sdµ(w)∈L^p−pq(D, α),

and this is equivalent to (x) via the substitutions=θ+_n+1^t . A consequence of these two theorems is that the property of being (p, q;α)-skew Carleson actually depends only on the quotientq/pand onα. We shall then introduce the following definition:

Definition 2.17. Letλ > 0 and γ >1−_n+1¹ . A finite positive Borel measureµ is (λ, γ)-skew Carleson if either

– λ≥1 and ˆµr0,λγ∈L^∞(D) for some (and hence all) r0∈(0,1); or, – λ <1 and ˆµr0,γ∈L^1−λ1 D,(n+ 1)(γ−1)

for some (and hence all) r0∈(0,1).

Thus Theorems 2.15 and 2.16 say thatµis (p, q;α)-skew Carleson if and only if it is (q/p, γ)-skew Carleson, whereα= (n+1)(γ−1). In particular, we shall writekµkq/p,γinstead ofkµkp,q;(n+1)(γ−1).

We end this section with the following easy (but useful) consequence of this definition:

Lemma 2.18. Let D⊂⊂Cⁿ be a bounded strongly pseudoconvex domain,λ >0andγ >1−_n+1¹ . Letµbe a(λ, γ)-skew Carleson measure, andβ > λ

n n+1 −γ

. Thenµβ=δ^(n+1)βµis a(λ, γ+^β_λ)- skew Carleson measure withkµβk_λ,γ+^β

λ ≈ kµkλ,γ.

Proof. First of all, remark that using Lemmas 2.1 and 2.2 it is easy to check that (µcβ)r≈δ^(n+1)βµˆr.

Assume 0< λ <1. By Theorem 2.16, we know that ˆµrδ−(n+1)(γ−1)λ∈L^1−λ1 (D). Therefore (µcβ)rδ^{−(n+1)(γ+βλ−1)λ}≈µˆrδ−(n+1)(γ−1)λ∈L^1−λ1 (D),

and again Theorem 2.16 implies thatµβ is (λ, γ+^β_λ)-skew Carleson withkµβk_λ,γ+^β

λ ≈ kµkλ,γ. Ifλ≥1, again Lemmas 2.1 and 2.2 yield

(µcβ)r,λγ+β ≈(µcβ)rδ−(n+1)(λγ+β−1)≈µˆrδ−(n+1)(λγ−1)≈µˆr,λγ

and Theorem 2.15 yields the assertion.

3. Proof of the main result

The proof of the main result will use two closely related operators. The first one is a Toeplitz-like operatorT_µ^β, depending on a parameterβ ∈N^∗ and on a finite positive Borel measureµ, defined by the formula

(3.1) T_µ^βf(z) =

Z

D

K(z, w)^βf(w)dµ(w)

for suitable functionsf:D→C; part of the work will exactly be identifying functional spaces that can act as domain and/or codomain of such an operator. We needβto be a natural number because the Bergman kernel in general might have zeroes andD is not necessarily simply connected.

The second operator S_t,µ^s,r depends onµ and three positive real parameters r, s, t >0 and is defined by

(3.2) S_µ,t^s,rf(z) =δ(z)^(n+1)s Z

D

|kz(w)|^t|f(w)|^rdµ(w),

again for suitable functionsf:D →C. This time the exponents do not need to be integers. Notice that Lemma 2.6 yields

(3.3) |S_µ,t^s,rf(z)| ≈δ(z)^(n+1)(s+2t) Z

D

|K(z, w)|^t|f(w)|^rdµ(w).

Therefore it is not surprising that, under suitable hypotheses we can use the norm of the operatorsS_t,µ^s,r to bound the norm of the operatorsT_µ^β. We start with a preliminary lemma:

Lemma 3.1. Let D ⊂⊂Cⁿ be a bounded strongly pseudoconvex domain, and µ a positive finite Borel measure onD. Then for every β≥t >0 we have

Z

D

|K(z, w)|^βdµ(w)δ(z)−(n+1)(β−t)Z

D

|K(z, w)|^tdµ(w). Proof. Givenz∈D putD1={w∈D| |K(z, w)| ≥1}andD0=D\D1. Then

Z

D

|K(z, w)|^βdµ(w) = Z

D0

|K(z, w)|^t|K(z, w)|^β−tdµ(w) + Z

D1

|K(z, w)|^t|K(z, w)|^β−tdµ(w)

≤ Z

D0

|K(z, w)|^tdµ(w) + sup

w∈D

|K(z, w)|^β−t Z

D1

|K(z, w)|^tdµ(w) sup

w∈D

|K(z, w)|^β−t Z

D

|K(z, w)|^tdµ(w),

and the assertion follows from the known estimate sup

w∈D

|K(z, w)| δ(z)⁻⁽ⁿ⁺¹⁾.

We then have the following estimates.

Lemma 3.2. Let D ⊂⊂Cⁿ be a bounded strongly pseudoconvex domain, and µ a positive finite Borel measure onD. Chooser≥1,s,t,p,q >0,σ,θ1>1−_n+1¹ andβ∈N^∗. Then:

(i) ifr= 1,q≥p,β≥tandθ1≤qh

σ

p +¹_q −¹_p+^t₂−β−si

we have

kT_µ^βfkp,(n+1)(σ−1) kS_µ,t^s,1fkq,(n+1)(θ1−1); (ii) ifr >1,q≥p/r,β ≥t/r, we have

kT_µ^βfkp,(n+1)(σ−1) kδ^{−(n+1)(γ−α2)}B^αµk^1/r1^′

1−λ,(n+1)(γ−1)kS_µ,t^s,rfk^1/r_{q,(n+1)(θ1−1)}, wherer^′ is the conjugate exponent ofrand

λ= 1 +r^′ 1

qr−1 p

<1, α=r^′

β−t r

, γ=r^′ λ

β+1

r

s− t 2

+θ1

qr −σ p

.

Proof. (i) Lemma 3.1, applied to the measure|f|µ, and (3.3) yield

|T_µ^βf(z)|^p≤ Z

D

|K(z, w)|^β|f(w)|dµ(w) p

δ(z)−(n+1)(β−t)pZ

D

|K(z, w)|^t|f(w)|dµ(w) p

δ(z)^{−(n+1)(β−t2+s)p}|S^s,1_µ,tf(z)|^p.

Therefore using H¨older’s inequality we obtain kT_µ^βfk^p_{p,(n+1)(σ−1)}

Z

D

|S_µ,t^s,1f(z)|^pδ(z)^{−(n+1)[(β−t2+s)p+1−σ]}dν(z)

Z

D

|S_µ,t^s,1f(z)|^qδ(z)^{−(n+1)[(β−t2+s)q+(1−pσ)q]}dν(z) p/q

=kS_µ,t^s,1fk^p_q,(n+1)q[σ−1

p +₂^t−β−s]

kS_µ,t^s,1fk^p_q,(n+1)(θ

1−1), where the last step follows from [3, Lemma 2.10].

(ii) Writing β = _r^t + _r^α_′, using H¨older’s inequality and recalling the definition of the Berezin transform we obtain

|T_µ^βf(z)|^p≤ Z

D

|K(z, w)|^t|f(w)|^rdµ(w) ^p/rZ

D

|K(z, w)|^αdµ(w) ^p/r′

Z

D

|K(z, w)|^t|f(w)|^rdµ(w) ^p/r

δ(z)^{−(n+1)αp2r′} |B^αµ(z)|^p/r′ .

Therefore, recalling thatα/r^′ =β−t/rand using again H¨older’s inequality, we have kT_µ^βfk^p_{p,(n+1)(σ−1)}

Z

D

Z

D

|K(z, w)|^t|f(w)|^rdµ(w) p/r

|B^αµ(z)|^p/r′δ(z)^{(n+1)(σ−1−αp2r′)}dν(z)

Z

D

|S_µ,t^s,rf(z)|^p/r|B^αµ(z)|^p/r′δ(z)^{(n+1)p[σ−1p −2r′α −(s+2t)1r]}dν(z)

≤ Z

D

|S_µ,t^s,rf(z)|^qδ(z)^{(n+1)(θ1−1)}dν(z)

p/qrZ

D

|B^αµ(z)|^{r′(qr−p)pqr} δ(z)^{(n+1)(τ−1)}dν(z) 1−^p

qr

=kS_µ,t^s,rfk^p/r_{q,(n+1)(θ1−1)}kδ^{−(n+1)(γ−α2)}B^αµk^p/r1^′

1−λ,(n+1)(γ−1), whereλandγ are as in the statement and

τ = r^′ 1−λ

σ p −θ1

qr− α 2r^′ −

s+ t

2 1

r

.

Corollary 3.3. Let D⊂⊂Cⁿ be a bounded strongly pseudoconvex domain, andµa positive finite Borel measure onD. Forr >1,s,t >0,p,˜ q >0,α >0,γ∈Randθ1>1−_n+1¹ assume that

β = t r +α

r^′ ∈N and λ= 1 +r

˜ p−1

q <1, wherer^′ =r/(r−1) is the conjugate exponent ofr. Then

kT_µ^βfkτ,(n+1)(σ−1) kδ^{−(n+1)(γ−α2)}B^αµk^1/r1^′

1−λ,(n+1)(γ−1)kS_µ,t^s,rfk^1/r_{q,(n+1)(θ1−1)}, where

τ= 1

1−λ+¹_p˜

and if σ >1−_n+1¹ , we have

(3.4) σ=τ

α−λγ r^′ + 1

qr

θ1+q

s+ t 2

.

Proof. The assertion is a consequence of Lemma 3.2.(ii) applied with p=τ. Indeed, first of all, sinceλ <1, we have ˜p > rq; from this it follows that

1−r

˜

p > 1−r

qr ⇐⇒ 1−λ+1

˜ p> 1

qr ⇐⇒ q > τ r as needed. Furthermore

1 +r^′ 1

qr− 1 τ

= 1 + r^′−1

q −r^′+r^′+rr^′

˜ p −r^′

q −r^′

˜

p = 1 +r

˜ p−1

q =λ and

r^′ λ

β+1

r

s−t 2

+θ1

qr−σ τ

=r^′ λ

β+1

r

s− t 2

+θ1

qr +λγ−α r^′ − 1

qr

θ1+q

s+ t 2

=γ . The mapping properties of the operatorsT_µ^β andS_t,µ^s,r can be used to give criteria for a measure µto be (λ, γ)-skew Carleson, which is particularly useful whenλ <1. We start withT_µ^β:

Proposition 3.4. Let D ⊂⊂ Cⁿ be a bounded strongly pseudoconvex domain, and µ a positive finite Borel measure onD. Take 0< q < p <∞,θ1,θ2>1−_n+1ⁿ andβ ∈Nsuch that

β > 1

pmax{1, θ1, p−1 +θ1}. Put

λ= 1 + 1 p−1

q <1 and γ= 1 λ

β+θ1

p −θ2

q

. Assume that T_µ^β is bounded from A^p D,(n+ 1)(θ1−1)

toA^q D,(n+ 1)(θ2−1)

, with operator normkT_µ^βk. Thenµis(λ, γ)-skew Carleson, and

kδ^{−(n+1)(γ−α2)}B^αµk ¹

1−λ,(n+1)(γ−1) kT_µ^βk for allα > λγ+_n+1ⁿ (1−λ).

Proof. Let {ak} be an r-lattice in D, and {rk} a sequence of Rademacher functions (see [10, Appendix A]). Set

τ= (n+ 1) β

2 −θ1

p

,

and, for everya∈D, putfa =δ(a)^τk^β_a. Then Lemma 2.10 implies that ft=

X∞

k=0

ckrk(t)fak

belongs toA^p D,(n+ 1)(θ1−1)

for allc={ck} ∈ℓ^p, andkfkp,(n+1)(θ1−1) kckp.

Since, by assumption,T_µ^βis bounded fromA^p D,(n+ 1)(θ1−1)

toA^q D,(n+ 1)(θ2−1) we have

kT_µ^βftk^q_{q,(n+1)(θ2−1)}= Z

D

X∞

k=0

ckrk(t)T_µ^βfak(z)

q

δ(z)^{(n+1)(θ2−1)}dν(z)

≤ kT_µ^βk^qkftk^q_{p,(n+1)(θ1−1)} kT_µ^βk^qkck^q_p.

Integrating both sides on [0,1] with respect to t and using Khinchine’s inequality (see, e.g., [19]) we obtain

Z

D

X∞

k=0

|ck|²|T_µ^βfak(z)|²

!^q/2

δ(z)^{(n+1)(θ2−1)}dν(z) kT_µ^βk^qkck^q_p. SetBk =BD(ak, r). We have to consider two cases: q≥2 and 0< q <2.

Ifq≥2, using the fact thatkakq/2≤ kak1for everya∈ℓ¹ we get X∞

k=0

|ck|^q Z

Bk

|T_µ^βfak(z)|^qδ(z)^{(n+1)(θ2−1)}dν(z)

≤ Z

D

X∞

k=0

|ck|²|T_µ^βfak(z)|²χBk(z)

!^q/2

δ(z)^{(n+1)(θ2−1)}dν(z)

≤ Z

D

X∞ k=0

|ck|²|T_µ^βfak(z)|²

!q/2

δ(z)^{(n+1)(θ2−1)}dν(z). If instead 0< q <2, using H¨older’s inequality, we obtain

X∞

k=0

|ck|^q Z

Bk

|T_µ^βfak(z)|^qδ(z)^{(n+1)(θ2−1)}dν(z)

≤ Z

D

X∞ k=0

|ck|²|T_µ^βfak(z)|²

!^q_{2 ∞} X

k=0

χBk(z)

!1−^q₂

δ(z)^{(n+1)(θ2−1)}dν(z)

Z

D

X∞

k=0

|ck|²|T_µ^βfa_k(z)|²

!q/2

δ(z)^{(n+1)(θ2−1)}dν(z), where we used the fact that eachz∈D belongs to no more thanmof theBk.

Summing up, for anyq >0 we have X∞

k=0

|ck|^q Z

Bk

|T_µ^βfa_k(z)|^qδ(z)^{(n+1)(θ2−1)}dν(z) kT_µ^βk^qkck^q_p. Now Lemmas 2.1, 2.2 and 2.5 (see also [4, Corollary 2.7]) yield

|T_µ^βfak(ak)|^q δ(ak)^−(n+1)θ2 Z

Bk

|T_µ^βfak(z)|^qδ(z)^{(n+1)(θ2−1)}dν(z), and so we get

X∞

k=0

|ck|^qδ(ak)^(n+1)θ2|T_µ^βfa_k(ak)|^q kT_µ^βk^qkck^q_p.

On the other hand, using Lemmas 2.6 and 2.7, we obtain T_µ^βfak(ak) =δ(ak)^τ

Z

D

K(ak, w)^βkak(w)^βdµ(w)δ(ak)^τ+(n+1)β2 Z

D

|K(ak, w)|^2βdµ(w)

≥δ(ak)^τ+(n+1)β2 Z

BD(a_k,r)

|K(ak, w)|^2βdµ(w) δ(ak)^{τ−(n+1)3β2}µ BD(ak, r)

=δ(ak)^{−(n+1)[β+θp1]}µ BD(ak, r) . Putting all together we get

X∞

k=0

|ck|^q µ BD(ak, r) δ(ak)^(n+1)λγ

!^q

kT_µ^βk^qkck^q_p.

Setd={dk}, where

dk =µ BD(ak, r) δ(ak)^(n+1)λγ .

Then by duality we get {d^q_k} ∈ ℓ^p/(p−q) with k{d^q_k}kp/(p−q) kT_µ^βk^q, because p/(p−q) is the conjugate exponent ofp/q >1. This means that d∈ℓ^pq/(p−q)=ℓ^1/(1−λ)with

kdk ¹

1−λ kT_µ^βk. Since

dk ≈µˆr(ak)δ(ak)(n+1)(1−λγ),

the assertion then follows from Theorem 2.16.

Remark 3.5. Note that a similar result holds also for λ≥1 and can be strengthened to give yet another characterization of skew Carleson measures. Since such result is not needed in the present paper, we prefer to omit it here, and to present it in a forthcoming paper.

We can now prove a technical result involving the operators S_µ,t^s,r that will be crucial for the proof of our main theorem.

Proposition 3.6. Let D ⊂⊂ Cⁿ be a bounded strongly pseudoconvex domain, and µ a posi- tive finite Borel measure on D. Fix q > 1, p > 0, θ1, θ2 > 1 − _n+1¹ and r, s > 0, and t > ¹_pmax{1, θ2, p−1 +θ2}>0. Set

λ= 1 + r p−1

q and γ= 1

λ t

2+θ2r p −θ1

q −s

. Assume thatλ >0 andγ >1−_n+1¹ , and that there exists K >0such that (3.5) kS_µ,t^s,rfk_q,(n+1)(θ1−1)≤Kkfk^r_{p,(n+1)(θ2−1)} for allf ∈A^p D,(n+ 1)(θ2−1)

. Thenµ is a(λ, γ)-skew Carleson measure with kµkλ,µK.

Proof. Let us first consider the caseλ≥1. Givena∈D andσ∈Nsuch that pσ > θ2,

set

f_a^σ(z) =ka(z)^σ, forz∈D. By Proposition 2.9 we have

(3.6) kf_a^σk^r_{p,(n+1)(θ2−1)}δ(a)^{(n+1)(θ2rp−rσ2)}.

Now fix ρ >0. Clearly, there is a ˆρ >0 depending only onρ such thatz, w∈BD(a, ρ) implies w∈BD(z,ρ) for allˆ a∈D. By Lemma 2.2 we can findδ1>0 such that ifδ(a)< δ1thenδ(z)< δρˆ

for all z ∈BD(a, ρ), where δρˆ>0 is given by Lemma 2.7. Then ifδ(a)< δ1 using Lemmas 2.1, 2.2 and 2.7 we have

kS_µ,t^s,rf_a^σk^q_{q,(n+1)(θ1−1)}= Z

D

|S_µ,t^s,rf_a^σ(z)|^qδ(z)^{(n+1)(θ1−1)}dν(z)

≥ Z

BD(a,ρ)

|S_µ,t^s,rf_a^σ(z)|^qδ(z)^{(n+1)(θ1−1)}dν(z) δ(a)^{(n+1)(θ1−1)}

Z

BD(a,ρ)

δ(z)^(n+1)qs Z

D

|kz(w)|^t|f_a^σ(w)|^rdµ(w) q

dν(z)

δ(a)^{(n+1)(θ1−1+qs)} Z

BD(a,ρ)

"Z

BD(a,ρ)

|kz(w)|^t|f_a^σ(w)|^rdµ(w)

#q

dν(z)

δ(a)^{(n+1)(θ1−1+qs−12σrq)} Z

BD(a,ρ)

δ(z)^{n+12 tq}

"Z

BD(a,ρ)

|K(z, w)|^tdµ(w)

#q

dν(z) δ(a)^{(n+1)(θ1−1+qs−12σrq+12tq)}

Z

BD(a,ρ)

δ(z)^−(n+1)tqµ BD(a, ρ)q

dν(z) δ(a)^{(n+1)(θ1+qs−12σrq−12tq)}µ BD(a, ρ)q

. Recalling (3.5) and (3.6), we get

µ BD(a, ρ)

Kδ(a)^{(n+1)(σr2+2t−s−θq1)}kf_a^σk^r_p,(n+1)(θ2−1)

Kδ(a)^{(n+1)(σr2+2t−s−θq1+θ2rp−rσ2)} Kν BD(a, ρ)^t2−s−^θ1

q+θ2r p .

Sinceµ is a finite measure, a similar estimate holds whenδ(a)≥δ1. Then Theorem 2.15 implies thatµis (λ, γ)-skew Carleson withkµkλ,γ Kas claimed.

Now let us assume 0< λ <1. Assume firstr= 1. Chooseβ ∈Nwith β≥tand set σ=θ1+q

β− t

2 +s

>1− 1 n+ 1 . We can apply (3.5) and Lemma 3.2.(i) with p=qto get

kT_µ^βfk_q,(n+1)(σ−1)Kkfk_p,(n+1)(θ2−1). Therefore Proposition 3.4 implies thatµis (λ,γ)-skew Carleson with˜

˜ γ= 1

λ

β+θ2

p −σ q

= 1 λ

t 2 +θ2

p −θ1

q −s

=γ , andkµkλ,γK as claimed.

Assume nowr >1, and chooseα >0 so that β= t

r+ α r^′ > 1

pmax{1, θ2, p−1 +θ2} andβ∈N. We also require that αis such thatα > λγ+_n+1ⁿ (1−λ) and

σ:=τ

α−λγ r^′ + 1

qr

θ1+q

s+ t 2

>1− 1 n+ 1 ,

where

τ= 1

1−λ+¹_p .

Assume for a moment that µ has compact support. Then kδ^{−(n+1)(γ−α2)}B^αµk ¹

1−λ,(n+1)(γ−1) is finite; therefore (3.5) and Corollary 3.3 applied with ˜p = p imply that T_µ^β is bounded from A^p D,(n+ 1)(θ2−1)

toA^τ D,(n+ 1)(σ−1) , with kT_µ^βk Kkδ^{−(n+1)(γ−α2)}B^αµk^1/r1^′

1−λ,(n+1)(γ−1).

Proposition 3.4 then yields thatµis (˜λ,γ)-skew Carleson with˜

˜λ= 1 +1 p−1

τ =λ and

˜ γ= 1

λ

β+θ2

p −σ τ

= 1 λ

β+θ2

p − α r^′ +λγ

r^′ −θ1

qr−1 r

s+t

2

= 1 λ

t 2r+θ2

p + t 2r^′ +θ2r

pr^′ − θ1

qr^′ − s r^′ −θ1

qr−s r

= 1 λ

t 2−θ1

q −s+θ2r p

=γ . Furthermore, we also have

kδ^{−(n+1)(γ−α2)}B^αµk ¹

1−λ,(n+1)(γ−1)Kkδ^{−(n+1)(γ−α2)}B^αµk^1/r1^′

1−λ,(n+1)(γ−1)

and thus

kδ^{−(n+1)(γ−α2)}B^αµk ¹

1−λ,(n+1)(γ−1)K .

An easy limit argument then shows that this holds even when the support of µis not compact, and then, by Theorem 2.16,µis (λ, γ)-skew Carleson withkµkλ,γK.

We are left with the case 0< r <1. ChooseR >1 and setµ^∗=δ^Aµ, with A= (n+ 1)(R−r)θ2

p .

First of all, fixr0∈(0,1) and set R0= ¹₂(1 +r0). Then, for anyz∈D, Lemmas 2.1, 2.2 and 2.5 yield

|f(z)|^p 1 ν BD(z, r0)

Z

BD(z,R0)

|f(w)|^pdν(w)δ(z)^−(n+1)θ2 Z

BD(z,R0)

|f(w)|^pδ(w)^{(n+1)(θ2−1)}dν(w)

≤δ(z)^−(n+1)θ2kfk^p_{p,(n+1)(θ2−1)}.