Skew Carleson Measures in Strongly Pseudoconvex Domains

Abstract

Given a bounded strongly pseudoconvex domain D in $\mathbb {C}^n$ with smooth boundary, we give a characterization through products of functions in weighted Bergman spaces of $(\lambda ,\gamma )$-skew Carleson measures on D, with $\lambda >0$ and $\gamma >1-\frac{1}{n+1}$.

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 $\Delta \subset \mathbb {C}$. Let A be a Banach space of holomorphic functions on a domain $D\subset \mathbb {C}^n$; given $p\ge 1$, a finite positive Borel measure $\mu$ on D is a Carleson measure of A (for p) if there is a continuous inclusion $A\hookrightarrow L^p(\mu )$, that is there exists a constant $C>0$ such that

$$\begin{aligned} \forall f\in A \quad \int _D |f|^p\,d\mu \le C\Vert f\Vert _A^p. \end{aligned}$$

In this paper, we are interested in Carleson measures for Bergman spaces, that is spaces of $L^p$ holomorphic functions, usually denoted by $A^p$ (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 [10] (see also Oleinik and Pavlov [21] and Oleinik [20]) for the Bergman spaces $A^p(\Delta )$, Cima and Wogen [8] in the case of the unit ball $B^n\subset \mathbb {C}^n$, Zhu [24] in the case of bounded symmetric domains, Cima and Mercer [7] for Bergman spaces in strongly pseudoconvex domains $A^p(D)$, and Luecking [18] for more general domains.

Given $D\subset \subset \mathbb {C}^n$ a bounded strongly pseudoconvex domain in $\mathbb {C}^n$ with smooth $C^\infty$ boundary, a positive finite Borel measure $\mu$ on D and $0<p<+\infty$, we denote by $L^p(\mu )$ the set of complex-valued $\mu$-measurable functions $f:D\rightarrow \mathbb {C}$ such that

$$\begin{aligned} \Vert f\Vert _{p,\mu }:=\left[ \int _D |f(z)|^p\,d\mu (z)\right] ^{1/p}<+\infty . \end{aligned}$$

If $\mu =\delta ^\alpha \nu$ for some $\alpha \in \mathbb {R}$, where $\delta (z)=d(z,\partial D)$ is the distance from the boundary of D and $\nu$ is the Lebesgue measure, the weighted Bergman space is defined as

$$\begin{aligned} A^p(D,\alpha )=L^p(\delta ^\alpha \nu )\cap \mathcal {O}(D), \end{aligned}$$

where $\mathcal {O}(D)$ 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 $A^p(D, \alpha )$ into $L^q$ spaces. Given, $0<p$, $q<+\infty$ and $\alpha >-1$, a finite positive Borel measure $\mu$ is called a $(p,q;\alpha )$-skew Carleson measure if $A^p(D,\alpha )\hookrightarrow L^q(\mu )$ continuously, that is there exists a constant $C>0$ such that

$$\begin{aligned} \int _D |f(z)|^q\,d\mu (z)\le C \Vert f\Vert _{p,\alpha }^q \end{aligned}$$

for all $f\in A^p(D,\alpha )$. Investigation on $(p,q;\alpha )$-skew Carleson measure has been started by Luecking in [19] and recently extended by Hu et al. in [13], where these measures are called $(p,q,\alpha )$ Bergman Carleson measures. It turns out (see [13] and the next section for details) that the property of being $(p,q;\alpha )$-skew Carleson depends only on the quotient q / p and on $\alpha$, allowing us to define $(\lambda ,\gamma )$-skew Carleson measures for $\lambda >0$ and $\gamma >1-\frac{1}{n+1}$. Roughly speaking, a measure is $(\lambda ,\gamma )$-skew Carleson if and only if it is a $(p,q;(n+1)(\gamma -1))$-skew Carleson measure for some (and hence any) p, q such that $q/p =\lambda$ (see Definition 2.17).

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

Theorem 1.1

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, and let $\mu$ be a positive finite Borel measure on D. Fix an integer $k\ge 1$, and let $0<p_j$, $q_j<\infty$ and $1-\frac{1}{n+1}<\theta _j$ be given, for $j=1,\ldots , k$. Set

$$\begin{aligned} \lambda =\sum _{j=1}^k\frac{q_j}{p_j}\quad and \quad \gamma =\frac{1}{\lambda }\sum _{j=1}^k \theta _j\frac{q_j}{p_j}. \end{aligned}$$

Then $\mu$ is a $(\lambda ,\gamma )$-skew Carleson measure if and only if there exists $C>0$ such that

$$\begin{aligned} \int _D \prod _{j=1}^k |f_j(z)|^{q_j}\,d\mu (z)\le C \prod _{j=1}^k \Vert f_j\Vert _{p_j,(n+1)(\theta _j-1)}^{q_j} \end{aligned}$$

(1.1)

for any $f_j\in A^{p_j}\bigl (D,(n+1)(\theta _j-1)\bigr )$.

This result generalizes the analogue one obtained by Pau and Zhao in [22] on the unit ball of $\mathbb {C}^n$. The proof relies on the properties of two closely related operators. The first one is a Toeplitz-like operator $T^\beta _\mu$ [see (3.1)], depending on a parameter $\beta \in \mathbb {N}^*$ and on a finite positive Borel measure $\mu$, and the main issue consists in identifying functional spaces that can act as domain and/or codomain of such an operator. The second operator, $S^{s,r}_{t,\mu }$ [see (3.2)], depends on $\mu$ and three positive real parameters r, s, $t>0$, and its norm can be used to bound the norm of the operators $T^\beta _\mu$, under suitable assumptions. In particular, the key step in the proof of the necessity implication in the case $0<\lambda <1$ consists in finding criteria for a measure to be $(\lambda ,\gamma )$-skew Carleson. These criteria are expressed in terms of mapping properties of the two operators $T^\beta _\mu$ and $S^{s,r}_{t,\mu }$ in the technical Propositions 3.4 and 3.6.

The paper is structured as follows. In Sect. 2 we shall collect the preliminary results and definitions. In Sect. 3 we shall study the properties of the operators $T^\beta _\mu$ and $S^{s,r}_{t,\mu }$ 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, $D\subset \subset \mathbb {C}^n$ will be a bounded strongly pseudoconvex domain in $\mathbb {C}^n$ with smooth $C^\infty$ boundary. Furthermore, we shall use the following notations:

• $\delta :D\rightarrow \mathbb {R}^+$ will denote the Euclidean distance from the boundary of D, that is $\delta (z)=d(z,\partial D)$;

• given two non-negative functions f,  $g:D\rightarrow \mathbb {R}^+$ we shall write $f\preceq g$ to say that there is $C>0$ such that $f(z)\le C g(z)$ for all $z\in D$. The constant C is independent of $z\in D$, but it might depend on other parameters (r, $\theta$, etc.);

• given two strictly positive functions f,  $g:D\rightarrow \mathbb {R}^+$ we shall write $f\approx g$ if $f\preceq g$ and $g\preceq f$, that is if there is $C>0$ such that $C^{-1} g(z)\le f(z)\le C g(z)$ for all $z\in D$;

• $\nu$ will be the Lebesgue measure;

• $\mathcal {O}(D)$ will denote the space of holomorphic functions on D, endowed with the topology of uniform convergence on compact subsets;

• given $0< p< +\infty$, the Bergman space$A^p(D)$ is the (Banach if $p\ge 1$) space $L^p(D)\cap \mathcal {O}(D)$, endowed with the $L^p$-norm;

• more generally, if $\mu$ is a positive finite Borel measure on D and $0<p<+\infty$ we shall denote by $L^p(\mu )$ the set of complex-valued $\mu$-measurable functions $f:D\rightarrow \mathbb {C}$ such that

  $$\begin{aligned} \Vert f\Vert _{p,\mu }:=\left[ \int _D |f(z)|^p\,d\mu (z)\right] ^{1/p}<+\infty . \end{aligned}$$

  If $\mu =\delta ^\beta \nu$ for some $\beta \in \mathbb {R}$, we shall denote by $A^p(D,\beta )$ the weighted Bergman space

  $$\begin{aligned} A^p(D,\beta )=L^p(\delta ^\beta \nu )\cap \mathcal {O}(D), \end{aligned}$$

  and we shall write $\Vert \cdot \Vert _{p,\beta }$ instead of $\Vert \cdot \Vert _{p,\delta ^\beta \nu }$;

• $K:D\times D\rightarrow \mathbb {C}$ will be the Bergman kernel of D;

• for each $z_0\in D$ we shall denote by $k_{z_0}:D\rightarrow \mathbb {C}$ the normalized Bergman kernel defined by

  $$\begin{aligned} k_{z_0}(z)=\frac{K(z,z_0)}{\sqrt{K(z_0,z_0)}}=\frac{K(z,z_0)}{\Vert K(\cdot ,z_0)\Vert _2}; \end{aligned}$$

• given $r\in (0,1)$ and $z_0\in D$, we shall denote by $B_D(z_0,r)$ the Kobayashi ball of center $z_0$ and radius $\frac{1}{2}\log \frac{1+r}{1-r}$.

We refer to, e.g., [1, 2, 14, 15], for definitions, basic properties and applications to geometric function theory of the Kobayashi distance; and to [11, 12, 16, 23] 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

([17, Corollary 7], [4, Lemma 2.1]) Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, and $r\in (0,1)$. Then

$$\begin{aligned} \nu \bigl (B_D(\cdot ,r)\bigr )\approx \delta ^{n+1}, \end{aligned}$$

(where the constant depends on r).

Lemma 2.2

([4, Lemma 2.2]) Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain. Then there is $C>0$ such that

$$\begin{aligned} \frac{1-r}{C}\delta (z_0)\le \delta (z) \le \frac{C}{1-r}\delta (z_0) \end{aligned}$$

for all $r\in (0,1)$, $z_0\in D$ and $z\in B_D(z_0,r)$.

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

Definition 2.3

Let $D\subset \subset \mathbb {C}^n$ be a bounded domain, and $r>0$. A r-lattice in D is a sequence $\{a_k\}\subset D$ such that $D=\bigcup _{k} B_D(a_k,r)$ and there exists $m>0$ such that any point in D belongs to at most m balls of the form $B_D(a_k,R)$, where $R=\frac{1}{2}(1+r)$.

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

Lemma 2.4

([4, Lemma 2.5]) Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain. Then for every $r\in (0,1)$ there exists an r-lattice in D, that is there exist $m\in \mathbb {N}$ and a sequence $\{a_k\}\subset D$ of points such that $D=\bigcup _{k=0}^\infty B_D(a_k,r)$ and no point of D belongs to more than m of the balls $B_D(a_k,R)$, where $R={\frac{1}{2}}(1+r)$.

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

Lemma 2.5

([4, Corollary 2.8]) Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain. Given $r\in (0,1)$, set $R={\frac{1}{2}}(1+r)\in (0,1)$. Then there exists a constant $K_r>0$ depending on r such that

$$\begin{aligned} \forall {z_0\in D \quad \forall z\in B_D(z_0,r)}\ \ \ \ \chi (z)\le {\frac{K_r}{\nu \left( B_D(z_0,r)\right) }}\int _{B_D(z_0,R)}\chi \,d\nu \end{aligned}$$

for every nonnegative plurisubharmonic function $\chi :D\rightarrow \mathbb {R}^+$.

We shall also need a few estimates on the behavior of the Bergman kernel. The first one is classical (see, e.g., [11]):

Lemma 2.6

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain. Then

$$\begin{aligned} \Vert K(\cdot ,z_0)\Vert _2=\sqrt{K(z_0,z_0)}\approx \delta (z_0)^{-(n+1)/2}\quad \hbox {and}\quad \Vert k_{z_0}\Vert _2\equiv 1 \end{aligned}$$

for all $z_0\in D$.

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

Lemma 2.7

([17, Theorem 12], [4, Lemma 3.2 and Corollary 3.3]) Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain. Then for every $r\in (0,1)$ there exist $c_r>0$ and $\delta _r>0$ such that if $z_0\in D$ satisfies $\delta (z_0)<\delta _r$ then

$$\begin{aligned} \frac{c_r}{\delta (z_0)^{n+1}}\le |K(z,z_0)|\le \frac{1}{c_r\delta (z_0)^{n+1}} \end{aligned}$$

and

$$\begin{aligned} \frac{c_r}{\delta (z_0)^{n+1}}\le |k_{z_0}(z)|^2\le \frac{1}{c_r\delta (z_0)^{n+1}} \end{aligned}$$

for all $z\in B_D(z_0,r)$.

Remark 2.8

Note that in the previous lemma the estimates from above hold even when $\delta (z_0)\ge \delta _r$, possibly with a different constant $c_r$. Indeed, when $\delta (z_0)\ge \delta _r$ and $z\in B_D(z_0,r)$ by Lemma 2.2 there is $\tilde{\delta }_r>0$ such that $\delta (z)\ge \tilde{\delta }_r$; as a consequence we can find $M_r>0$ such that $|K(z,z_0)|\le M_r$ as soon as $\delta (z_0)\ge \delta _r$ and $z\in B_D(z_0,r)$, and the assertion follows from the fact that D is a bounded domain.

A very useful integral estimate is the following:

Proposition 2.9

([17, Corollary 11, Theorem 13], [3, Theorem 2.7]) Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, and $z_0\in D$. Let $0<p<+\infty$ and $-1<\beta <(n+1)(p-1)$. Then

$$\begin{aligned} \int _D |K(z,w)|^p\delta (w)^\beta \,d\nu (w)\preceq \delta (z)^{\beta -(n+1)(p-1)} \end{aligned}$$

and

$$\begin{aligned} \int _D |k_z(w)|^p\delta (w)^\beta \,d\nu (w)\preceq \delta (z)^{\beta -(n+1)(\frac{p}{2}-1)}. \end{aligned}$$

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

Lemma 2.10

([13, Lemma 2.6]) Let $p>0$ and $\theta >1-\frac{1}{n+1}$ be given, and let $\alpha =(n+1)(\theta -1)>-1$. Take $\beta \in {\mathbb {N}}$ such that $\beta p>\max \{\theta ,(p-1)\frac{n}{n+1}+\theta \}$ and put

$$\begin{aligned} \tau =(n+1)\left[ \frac{\beta }{2}-\frac{\theta }{p}\right] . \end{aligned}$$

For each $a\in D$ set $f_a=\delta (a)^\tau k_a^\beta$. Let $\{a_k\}$ be an r-lattice and $\mathbf {c}=\{c_k\}\in \ell ^p$, and put

$$\begin{aligned} f=\sum _{k=0}^\infty c_kf_{a_k}. \end{aligned}$$

Then $f\in A^p(D,\alpha )$ with $\Vert f\Vert _{p,\alpha }\preceq \Vert \mathbf {c}\Vert _p$.

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

Definition 2.11

Let $0<p$, $q<+\infty$ and $\alpha >-1$. A $(p,q;\alpha )$-skew Carleson measure is a finite positive Borel measure $\mu$ such that

$$\begin{aligned} \int _D |f(z)|^q\,d\mu (z)\preceq \Vert f\Vert _{p,\alpha }^q \end{aligned}$$

for all $f\in A^p(D,\alpha )$. In other words, $\mu$ is $(p,q;\alpha )$-skew Carleson if $A^p(D,\alpha )\hookrightarrow L^q(\mu )$ continuously. In this case we shall denote by $\Vert \mu \Vert _{p,q;\alpha }$ the operator norm of the inclusion $A^p(D,\alpha )\hookrightarrow L^q(\mu )$.

Remark 2.12

When $p=q$ we recover the usual (non-skew) notion of Carleson measure for $A^p(D,\alpha )$.

Definition 2.13

Let $\theta \in \mathbb {R}$, and let $\mu$ be a finite positive Borel measure on D. Given $r\in (0,1)$, let ${\hat{\mu }}_{r,\theta }:D\rightarrow \mathbb {R}$ be defined by

$$\begin{aligned} {\hat{\mu }}_{r,\theta }(z)=\frac{\mu \bigl (B_D(z,r)\bigr )}{\nu \bigl (B_D(z,r)\bigr )^\theta }; \end{aligned}$$

we shall write ${\hat{\mu }}_r$ for ${\hat{\mu }}_{r,1}$.

We say that $\mu$ is a geometric $\theta$-Carleson measure if ${\hat{\mu }}_{r,\theta }\in L^\infty (D)$ for all $r\in (0,1)$, that is if for every $r>0$ we have

$$\begin{aligned} \mu \bigl (B_D(z,r)\bigr )\preceq \nu \bigl (B_D(z,r)\bigr )^\theta \end{aligned}$$

for all $z\in D$, where the constant depends only on r.

Notice that Lemma 2.1 yields

$$\begin{aligned} {\hat{\mu }}_{r,\theta }\approx \delta ^{-(n+1)(\theta -1)}{\hat{\mu }}_r. \end{aligned}$$

(2.1)

In [3] we proved (among other things) that, if $p\ge 1$, a measure $\mu$ is $(p,p;\alpha )$-skew Carleson if and only if it is geometric $\theta$-Carleson, where $\alpha =(n+1)(\theta -1)$. Hu, Lv and Zhu in [13] have given a similar geometric characterization of $(p,q;\alpha )$-skew Carleson measures for all values of p and q; to state their results we need another definition.

Definition 2.14

Let $\mu$ be a finite positive Borel measure on D, and $s>0$. The Berezin transform of level s of $\mu$ is the function $B^s\mu :D\rightarrow \mathbb {R}^+\cup \{+\infty \}$ given by

$$\begin{aligned} B^s\mu (z)=\int _D |k_z(w)|^s\,d\mu (w). \end{aligned}$$

The geometric characterization of $(p,q;\alpha )$-skew Carleson measures is different according to whether $p\le q$ or $p>q$. We first state the characterization for the case $p\le q$.

Theorem 2.15

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain. Let $0< p\le q<+\infty$ and $1-\frac{1}{n+1}<\theta$; set $\alpha =(n+1)(\theta -1)>-1$. Then the following assertions are equivalent:

• (i) $\mu$ is a $(p,q;\alpha )$-skew Carleson measure;

• (ii) $\mu$ is a geometric $\frac{q}{p}\theta$-Carleson measure;

• (iii) there exists $r_0\in (0,1)$ such that ${\hat{\mu }}_{r_0,\frac{q}{p}\theta }\in L^\infty (D)$;

• (iv) for every $r\in (0,1)$ and for every r-lattice $\{a_k\}$ in D we have

  $$\begin{aligned} \mu \bigl (B_D(a_k,r)\bigr )\preceq \nu \bigl (B_D(a_k,r)\bigr )^{\frac{q}{p}\theta }; \end{aligned}$$

• (v) there exists $r_0\in (0,1)$ and a $r_0$-lattice $\{a_k\}$ in D such that

  $$\begin{aligned} \mu \bigl (B_D(a_k,r_0)\bigr )\preceq \nu \bigl (B_D(a_k,r_0)\bigr )^{\frac{q}{p}\theta }; \end{aligned}$$

• (vi) for some (and hence all) $s>\theta \frac{q}{p}$ we have

  $$\begin{aligned} B^s\mu (a) \preceq \delta (a)^{(n+1)\left( \theta \frac{q}{p}-\frac{s}{2}\right) }; \end{aligned}$$

• (vii) there exists $C>0$ such that for some (and hence all) $t>0$ we have

  $$\begin{aligned} \int _D |K(z,a)|^{\theta \frac{q}{p}+\frac{t}{n+1}}\,d\mu (z)\preceq \delta (a)^{-t}. \end{aligned}$$

Moreover we have

$$\begin{aligned} \Vert \mu \Vert _{p,q;\alpha } \approx \left\| {\hat{\mu }}_{r,\frac{q}{p}\theta }\right\| _\infty \approx \left\| \delta ^{-(n+1)(\frac{q}{p}\theta -1)}{\hat{\mu }}_r\right\| _\infty \approx \left\| \delta ^{(n+1)\left( \frac{s}{2}-\theta \frac{q}{p}\right) }B^s\mu \right\| _\infty . \end{aligned}$$

(2.2)

Proof

The equivalence of (i)–(vi), as well as the equivalence for the norms, follows from [13, Theorem 3.1] (and the equivalence of (ii)–(v) was already in [3]).

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

$$\begin{aligned} \int _D |K(z,a)|^s\,d\mu (z)\preceq \delta (a)^{(n+1)\left( \theta \frac{q}{p}-s\right) }. \end{aligned}$$

Setting $t=(n+1)\left( s-\theta \frac{q}{p}\right)$, which is positive if and only if $s>\theta \frac{q}{p}$, we see that (vi) is equivalent to

$$\begin{aligned} \int _D |K(z,a)|^{\theta \frac{q}{p}+\frac{t}{n+1}}\,d\mu (z)\preceq \delta (a)^{-t}, \end{aligned}$$

that is to (vii). $\square$

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

Theorem 2.16

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain. Let $0< q< p<+\infty$ and $1-\frac{1}{n+1}<\theta$; put $\alpha =(n+1)(\theta -1)>-1$. Then the following assertions are equivalent:

• (i) $\mu$ is a $(p,q;\alpha )$-skew Carleson measure;

• (ii) ${\hat{\mu }}_r \delta ^{-\alpha \frac{q}{p}}\in L^{\frac{p}{p-q}}(D)$ for some (and hence any) $r\in (0,1)$;

• (iii) ${\hat{\mu }}_{r,\theta }\in L^{\frac{p}{p-q}}(D,\alpha )$ for some (and hence any) $r\in (0,1)$;

• (iv) ${\hat{\mu }}_{r,\theta \frac{q}{p}}\in L^{\frac{p}{p-q}}\bigl (D,-(n+1)\bigr )$ for some (and hence any) $r\in (0,1)$;

• (v) for some (and hence any) $r\in (0,1)$ and for some (and hence any) r-lattice $\{a_k\}$ in D we have $\{{\hat{\mu }}_{r,\theta \frac{q}{p}}(a_k)\}\in \ell ^{\frac{p}{p-q}}$;

• (vi) for some (and hence any) $r\in (0,1)$ and for some (and hence any) r-lattice $\{a_k\}$ in D we have $\{{\hat{\mu }}_{r}(a_k)\delta (a_k)^{(n+1)\left( 1-\theta \frac{q}{p} \right) }\}\in \ell ^{\frac{p}{p-q}}$;

• (vii) for some (and hence all) $s>\theta \frac{q}{p}+\frac{n}{n+1}\left( 1-\frac{q}{p}\right)$ we have

  $$\begin{aligned} \delta ^{-(n+1)\left( \theta \frac{q}{p}-\frac{s}{2}\right) }B^s\mu \in L^{\frac{p}{p-q}}\bigl (D,-(n+1)\bigr ); \end{aligned}$$

• (viii) for some (and hence all) $s>\theta \frac{q}{p}+\frac{n}{n+1}\left( 1-\frac{q}{p}\right)$ we have

  $$\begin{aligned} \delta ^{-(n+1)\left( \theta -\frac{s}{2}\right) }B^s\mu \in L^{\frac{p}{p-q}}(D,\alpha ); \end{aligned}$$

• (ix) for some (and hence all) $s>\theta \frac{q}{p}+\frac{n}{n+1}\left( 1-\frac{q}{p}\right)$ we have

  $$\begin{aligned} \delta ^{-(n+1)\left( \theta \frac{q}{p}-\frac{s}{2}+\frac{p-q}{p}\right) }B^s\mu \in L^{\frac{p}{p-q}}(D); \end{aligned}$$

• (x) for some (and hence all) $t>(n+1)\left( 1-\frac{q}{p}\right) \left( \frac{n}{n+1}-\theta \right)$ we have

  $$\begin{aligned} \delta ^t \int _D |K(\cdot ,w)|^{\theta +\frac{t}{n+1}}\,d\mu (w)\in L^{\frac{p}{p-q}}(D,\alpha ). \end{aligned}$$

  Moreover we have

  $$\begin{aligned} \Vert \mu \Vert _{p,q;\alpha } \approx \left\| \delta ^{-(n+1)(\theta -\frac{s}{2})}B^s\mu \right\| _{\frac{p}{p-q},\alpha } \approx \left\| \delta ^{-(n+1)(\theta -1)\frac{q}{p}}{\hat{\mu }}_r \right\| _{\frac{p}{p-q}} \end{aligned}$$

  (2.3)

Proof

The equivalence of (i), (ii), (vi) and (ix), as well as the equivalence of the norms, is in [13, Theorem 3.3].

Recalling that, by Lemma 2.1, ${\hat{\mu }}_{r,\theta }\approx {\hat{\mu }}_r \delta ^{(n+1)(1-\theta )}$, it is easy to see that the equalities

$$\begin{aligned} \begin{aligned} -\,(n+1)(\theta -1)\frac{q}{p}\frac{p}{p-q}&=(n+1)(1-\theta ) \frac{p}{p-q}+(n+1)(\theta -1)\\&=(n+1)\left( 1-\theta \frac{q}{p}\right) \frac{p}{p-q}-(n+1) \end{aligned} \end{aligned}$$

yield the equivalence of (ii), (iii) and (iv).

The fact that ${\hat{\mu }}_{r,\theta }\approx {\hat{\mu }}_r \delta ^{(n+1)(1-\theta )}$ immediately yields the equivalence between (v) and (vi).

The equalities

$$\begin{aligned} { \begin{aligned} -\,(n+1)\left( \theta \frac{q}{p}-\frac{s}{2}\right) \frac{p}{p-q}-(n+1)&=-\,(n+1)\left( \theta -\frac{s}{2}\right) \frac{p}{p-q}+(n+1)(\theta -1)\\&=-\,(n+1)\left( \theta \frac{q}{p}-\frac{s}{2}+\frac{p-q}{p}\right) \frac{p}{p-q} \end{aligned}}\end{aligned}$$

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

Finally, by Lemma 2.6, (viii) is equivalent to

$$\begin{aligned} \delta ^{-(n+1)(\theta -s)}\int _D |K(\cdot ,w)|^s\,d\mu (w)\in L^{\frac{p}{p-q}}(D,\alpha ), \end{aligned}$$

and this is equivalent to (x) via the substitution $s=\theta +\frac{t}{n+1}$. $\square$

A consequence of these two theorems is that the property of being $(p,q;\alpha )$-skew Carleson actually depends only on the quotient q / p and on $\alpha$. We shall then introduce the following definition:

Definition 2.17

Let $\lambda >0$ and $\gamma >1-\frac{1}{n+1}$. A finite positive Borel measure $\mu$ is $(\lambda ,\gamma )$-skew Carleson if either

• $\lambda \ge 1$ and ${\hat{\mu }}_{r_0,\lambda \gamma }\in L^\infty (D)$ for some (and hence all) $r_0\in (0,1)$; or,

• $\lambda <1$ and ${\hat{\mu }}_{r_0,\gamma }\in L^{\frac{1}{1-\lambda }}\bigl (D,(n+1)(\gamma -1)\bigr )$ for some (and hence all) $r_0\in (0,1)$.

Thus Theorems 2.15 and 2.16 say that $\mu$ is $(p,q;\alpha )$-skew Carleson if and only if it is $(q/p,\gamma )$-skew Carleson, where $\alpha =(n+1)(\gamma -1)$. In particular, we shall write $\Vert \mu \Vert _{q/p,\gamma }$ instead of $\Vert \mu \Vert _{p,q;(n+1)(\gamma -1)}$.

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

Lemma 2.18

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, $\lambda >0$ and $\gamma >1-\frac{1}{n+1}$. Let $\mu$ be a $(\lambda ,\gamma )$-skew Carleson measure, and $\beta >\lambda \left( \frac{n}{n+1}-\gamma \right)$. Then $\mu _\beta =\delta ^{(n+1)\beta }\mu$ is a $(\lambda , \gamma +\frac{\beta }{\lambda })$-skew Carleson measure with $\Vert \mu _\beta \Vert _{\lambda ,\gamma +\frac{\beta }{\lambda }}\approx \Vert \mu \Vert _{\lambda ,\gamma }$.

Proof

First of all, remark that using Lemmas 2.1 and 2.2 it is easy to check that

$$\begin{aligned} ({\widehat{\mu _\beta }})_r\approx \delta ^{(n+1)\beta }{\hat{\mu }}_r. \end{aligned}$$

Assume $0<\lambda <1$. By Theorem 2.16, we know that ${\hat{\mu }}_r\delta ^{-(n+1)(\gamma -1)\lambda }\in L^{\frac{1}{1-\lambda }}(D)$. Therefore

$$\begin{aligned} ({\widehat{\mu _\beta }})_r\delta ^{-(n+1)\big (\gamma +\frac{\beta }{\lambda }-1\big )\lambda }\approx {\hat{\mu }}_r\delta ^{-(n+1)(\gamma -1)\lambda }\in L^{\frac{1}{1-\lambda }}(D), \end{aligned}$$

and again Theorem 2.16 implies that $\mu _\beta$ is $(\lambda ,\gamma +\frac{\beta }{\lambda })$-skew Carleson with $\Vert \mu _\beta \Vert _{\lambda ,\gamma +\frac{\beta }{\lambda }}\approx \Vert \mu \Vert _{\lambda ,\gamma }$.

If $\lambda \ge 1$, again Lemmas 2.1 and 2.2 yield

$$\begin{aligned} ({\widehat{\mu _\beta }})_{r,\lambda \gamma +\beta }\approx ({\widehat{\mu _\beta }})_r\delta ^{-(n+1)(\lambda \gamma +\beta -1)}\approx {\hat{\mu }}_r \delta ^{-(n+1)(\lambda \gamma -1)}\approx {\hat{\mu }}_{r,\lambda \gamma } \end{aligned}$$

and Theorem 2.15 yields the assertion. $\square$

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 operator $T^\beta _\mu$, depending on a parameter $\beta \in \mathbb {N}^*$ and on a finite positive Borel measure $\mu$, defined by the formula

$$\begin{aligned} T^\beta _\mu f(z)=\int _D K(z,w)^\beta f(w)\,d\mu (w) \end{aligned}$$

(3.1)

for suitable functions $f:D\rightarrow \mathbb {C}$; part of the work will exactly be identifying functional spaces that can act as domain and/or codomain of such an operator. We need $\beta$ to be a natural number because the Bergman kernel in general might have zeroes and D is not necessarily simply connected.

The second operator $S^{s,r}_{t,\mu }$ depends on $\mu$ and three positive real parameters r, s, $t>0$ and is defined by

$$\begin{aligned} S^{s,r}_{\mu ,t}f(z)=\delta (z)^{(n+1)s}\int _D |k_z(w)|^t |f(w)|^r\,d\mu (w), \end{aligned}$$

(3.2)

again for suitable functions $f:D\rightarrow \mathbb {C}$. This time the exponents do not need to be integers. Notice that Lemma 2.6 yields

$$\begin{aligned} \left| S^{s,r}_{\mu ,t}f(z)\right| \approx \delta (z)^{(n+1)(s+\frac{t}{2})}\int _D |K(z,w)|^t|f(w)|^r\,d\mu (w). \end{aligned}$$

(3.3)

Therefore it is not surprising that, under suitable hypotheses we can use the norm of the operators $S^{s,r}_{t,\mu }$ to bound the norm of the operators $T^\beta _\mu$. We start with a preliminary lemma:

Lemma 3.1

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, and $\mu$ a positive finite Borel measure on D. Then for every $\beta \ge t>0$ we have

$$\begin{aligned} \int _D |K(z,w)|^\beta \,d\mu (w)\preceq \delta (z)^{-(n+1)(\beta -t)}\int _D |K(z,w)|^t\,d\mu (w). \end{aligned}$$

Proof

Let $z\in D$. Then

$$\begin{aligned} \begin{aligned} \int _D |K(z,w)|^\beta \,d\mu (w) \le \sup _{w\in D}|K(z,w)|^{\beta -t}\int _{D}|K(z,w)|^t\,d\mu (w), \end{aligned} \end{aligned}$$

and the assertion follows from the known estimate

$$\begin{aligned} \sup _{w\in D}|K(z,w)|\preceq \delta (z)^{-(n+1)}. \end{aligned}$$

$\square$

We then have the following estimates.

Lemma 3.2

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, and $\mu$ a positive finite Borel measure on D. Choose $r\ge 1$, s, t, p, $q>0$, $\sigma$, $\theta _1>1-\frac{1}{n+1}$ and $\beta \in \mathbb {N}^*$. Then:

• (i) if $r=1$, $q\ge p$, $\beta \ge t$ and $\theta _1\le q\left[ \frac{\sigma }{p}+\frac{1}{q}-\frac{1}{p}+\frac{t}{2}-\beta -s\right]$ we have

  $$\begin{aligned} \left\| T^\beta _\mu f\right\| _{p,(n+1)(\sigma -1)}\preceq \left\| S^{s,1}_{\mu ,t}f\right\| _{q,(n+1)(\theta _1-1)}; \end{aligned}$$

• (ii) if $r>1$, $q\ge p/r$, $\beta \ge t/r$, we have

  $$\begin{aligned} \left\| T^\beta _\mu f\right\| _{p,(n+1)(\sigma -1)}\preceq \left\| \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )} B^\alpha \mu \right\| ^{1/r'}_{\frac{1}{1-\lambda },(n+1)(\gamma -1)}\left\| S^{s,r}_{\mu ,t}f\right\| ^{1/r}_{q,(n+1)(\theta _1-1)}, \end{aligned}$$

  where $r'$ is the conjugate exponent of r and

  $$\begin{aligned}&\lambda =1+r'\left[ \frac{1}{qr}-\frac{1}{p}\right] <1, \quad \alpha =r'\left( \beta -\frac{t}{r}\right) ,\\&\quad \gamma =\frac{r'}{\lambda }\left[ \beta +\frac{1}{r}\left( s-\frac{t}{2}\right) + \frac{\theta _1}{qr}-\frac{\sigma }{p}\right] . \end{aligned}$$

Proof

(i) Lemma 3.1, applied to the measure $|f|\mu$, and (3.3) yield

$$\begin{aligned} \begin{aligned} \big |T^\beta _\mu f(z)\big |^p&\le \left[ \int _D |K(z,w)|^\beta |f(w)|\,d\mu (w)\right] ^p\\&\preceq \delta (z)^{-(n+1)(\beta -t)p}\left[ \int _D |K(z,w)|^t |f(w)|\,d\mu (w)\right] ^p\\&\preceq \delta (z)^{-(n+1)(\beta -\frac{t}{2}+s)p}|S^{s,1}_{\mu ,t}f(z)|^p. \end{aligned} \end{aligned}$$

Therefore using Hölder’s inequality we obtain

$$\begin{aligned} \begin{aligned} \big \Vert T^\beta _\mu f\big \Vert ^p_{p,(n+1)(\sigma -1)}&\preceq \int _D \left| S^{s,1}_{\mu ,t}f(z)\right| ^p \delta (z)^{-(n+1)\big [\big (\beta -\frac{t}{2}+s\big )p+1-\sigma \big ]}\,d\nu (z)\\&\preceq \left[ \int _D \left| S^{s,1}_{\mu ,t}f(z)\right| ^q \delta (z)^{-(n+1)[\big (\beta -\frac{t}{2}+s\big )q+\frac{(1-\sigma )q}{p}]}\,d\nu (z)\right] ^{p/q}\\&=\left\| S^{s,1}_{\mu ,t}f\right\| ^p_{q,(n+1)q\left[ \frac{\sigma -1}{p}+\frac{t}{2}-\beta -s\right] }\\&\preceq \left\| S^{s,1}_{\mu ,t}f\right\| ^p_{q,(n+1)(\theta _1-1)}, \end{aligned} \end{aligned}$$

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

(ii) Writing $\beta =\frac{t}{r}+\frac{\alpha }{r'}$, using Hölder’s inequality and recalling the definition of the Berezin transform we obtain

$$\begin{aligned} \begin{aligned} \left| T^\beta _\mu f(z)\right| ^p&\le \left[ \int _D |K(z,w)|^t|f(w)|^r\,d\mu (w)\right] ^{p/r}\left[ \int _D |K(z,w)|^{\alpha }\,d\mu (w)\right] ^{p/r'}\\&\preceq \left[ \int _D |K(z,w)|^t|f(w)|^r\,d\mu (w)\right] ^{p/r} \delta (z)^{-\frac{(n+1)\alpha p}{2r'}}|B^{\alpha }\mu (z)|^{p/r'}. \end{aligned} \end{aligned}$$

Therefore, recalling that $\alpha /r'=\beta -t/r$ and using again Hölder’s inequality, we have

$$\begin{aligned} {\begin{aligned} \left\| T^\beta _\mu f\right\| ^p_{p,(n+1)(\sigma -1)}&\preceq \int _D \left[ \int _D |K(z,w)|^t|f(w)|^r\,d\mu (w)\right] ^{p/r}|B^\alpha \mu (z)|^{p/r'} \delta (z)^{(n+1)\big (\sigma -1-\frac{\alpha p}{2r'}\big )} \,d\nu (z)\\&\preceq \int _D \Big |S^{s,r}_{\mu , t}f(z)\Big |^{p/r}|B^\alpha \mu (z)|^{p/r'} \delta (z)^{(n+1)p\big [\frac{\sigma -1}{p}-\frac{\alpha }{2r'}-\big (s+\frac{t}{2}\big )\frac{1}{r}\big ]}\,d\nu (z)\\&\le \left[ \int _D \Big |S^{s,r}_{\mu ,t}f(z)\Big |^q\delta (z)^{(n+1)(\theta _1-1)}\, d\nu (z)\right] ^{p/qr}\\&\quad \times \left[ \int _D |B^\alpha \mu (z)|^{\frac{pqr}{r'(qr-p)}}\delta (z)^{(n+1)(\tau -1)}\,d\nu (z)\right] ^{1-\frac{p}{qr}}\\&=\Big \Vert S^{s,r}_{\mu ,t}f\Big \Vert ^{p/r}_{q,(n+1)(\theta _1-1)}\left\| \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )}B^\alpha \mu \right\| ^{p/r'}_{\frac{1}{1-\lambda },(n+1)(\gamma -1)}, \end{aligned}}\end{aligned}$$

where $\lambda$ and $\gamma$ are as in the statement and

$$\begin{aligned} \tau =\frac{r'}{1-\lambda }\left[ \frac{\sigma }{p}-\frac{\theta _1}{qr}-\frac{\alpha }{2r'}-\left( s+\frac{t}{2}\right) \frac{1}{r}\right] . \end{aligned}$$

$\square$

Corollary 3.3

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, and $\mu$ a positive finite Borel measure on D. For $r>1$, s, $t>0$, ${\tilde{p}}$, $q>0$, $\alpha >0$, $\gamma \in \mathbb {R}$ and $\theta _1>1-\frac{1}{n+1}$ assume that

$$\begin{aligned} \beta =\frac{t}{r}+\frac{\alpha }{r'}\in \mathbb {N}\quad and \quad \lambda =1+\frac{r}{{\tilde{p}}}-\frac{1}{q}<1, \end{aligned}$$

where $r'=r/(r-1)$ is the conjugate exponent of r. Then

$$\begin{aligned} \left\| T^\beta _\mu f\right\| _{\tau ,(n+1)(\sigma -1)}\preceq \left\| \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )}B^\alpha \mu \right\| ^{1/r'}_{\frac{1}{1-\lambda },(n+1)(\gamma -1)}\Big \Vert S^{s,r}_{\mu ,t}f\Big \Vert ^{1/r}_{q,(n+1)(\theta _1-1)}, \end{aligned}$$

where

$$\begin{aligned} \tau =\frac{1}{1-\lambda +\frac{1}{{\tilde{p}}}} \end{aligned}$$

and if $\sigma >1-\frac{1}{n+1}$, we have

$$\begin{aligned} \sigma = \tau \left[ \frac{\alpha -\lambda \gamma }{r'} + \frac{1}{q r}\left( \theta _1+ q\left( s+\frac{t}{2}\right) \right) \right] . \end{aligned}$$

(3.4)

Proof

The assertion is a consequence of Lemma 3.2.(ii) applied with $p=\tau$. Indeed, first of all, since $\lambda <1$, we have ${\tilde{p}}>rq$; from this it follows that

$$\begin{aligned} \frac{1-r}{{\tilde{p}}}>\frac{1-r}{qr}\quad \Longleftrightarrow \quad 1-\lambda +\frac{1}{{\tilde{p}}}>\frac{1}{qr} \quad \Longleftrightarrow \quad q>\frac{\tau }{r} \end{aligned}$$

as needed. Furthermore

$$\begin{aligned} 1+r'\left[ \frac{1}{qr}-\frac{1}{\tau }\right] =1+\frac{r'-1}{q}-r'+r'+ \frac{rr'}{{\tilde{p}}}-\frac{r'}{q}-\frac{r'}{{\tilde{p}}}=1+\frac{r}{{\tilde{p}}}-\frac{1}{q}=\lambda \end{aligned}$$

and

$$\begin{aligned}&\frac{r'}{\lambda }\left[ \beta +\frac{1}{r}\left( s-\frac{t}{2}\right) + \frac{\theta _1}{qr}-\frac{\sigma }{\tau }\right] \\&\quad =\frac{r'}{\lambda } \left[ \beta +\frac{1}{r}\left( s-\frac{t}{2}\right) +\frac{\theta _1}{qr}+ \frac{\lambda \gamma -\alpha }{r'}-\frac{1}{qr}\left( \theta _1+q \left( s+\frac{t}{2}\right) \right) \right] =\gamma . \end{aligned}$$

$\square$

The mapping properties of the operators $T^\beta _\mu$ and $S^{s,r}_{t,\mu }$ can be used to give criteria for a measure $\mu$ to be $(\lambda ,\gamma )$-skew Carleson, which is particularly useful when $\lambda <1$. We start with $T^\beta _\mu$:

Proposition 3.4

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, and $\mu$ a positive finite Borel measure on D. Take $0<q<p<\infty$, $\theta _1$, $\theta _2>1-\frac{n}{n+1}$ and $\beta \in \mathbb {N}$ such that

$$\begin{aligned} \beta >\frac{1}{p}\max \{1,\theta _1,p-1+\theta _1\}. \end{aligned}$$

Put

$$\begin{aligned} \lambda =1+\frac{1}{p}-\frac{1}{q}<1\quad and \quad \gamma =\frac{1}{\lambda }\left( \beta +\frac{\theta _1}{p}-\frac{\theta _2}{q}\right) . \end{aligned}$$

Assume that $T^\beta _\mu$ is bounded from $A^{p}\bigl (D,(n+1)(\theta _1-1)\bigr )$ to $A^{q}\bigl (D,(n+1)(\theta _2-1)\bigr )$, with operator norm $\Vert T^\beta _\mu \Vert$. Then $\mu$ is $(\lambda ,\gamma )$-skew Carleson, and

$$\begin{aligned} \left\| \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )}B^\alpha \mu \right\| _{\frac{1}{1-\lambda },(n+1)(\gamma -1)}\preceq \Vert T^\beta _\mu \Vert \end{aligned}$$

for all $\alpha >\lambda \gamma +\frac{n}{n+1}(1-\lambda )$.

Proof

Let $\{a_k\}$ be an r-lattice in D, and $\{r_k\}$ a sequence of Rademacher functions (see [9, Appendix A]). Set

$$\begin{aligned} \tau =(n+1)\left[ \frac{\beta }{2}-\frac{\theta _1}{p}\right] , \end{aligned}$$

and, for every $a\in D$, put $f_a=\delta (a)^\tau k_a^\beta$. Then Lemma 2.10 implies that

$$\begin{aligned} f_t=\sum _{k=0}^\infty c_kr_k(t)f_{a_k} \end{aligned}$$

belongs to $A^{p}\bigl (D,(n+1)(\theta _1-1)\bigr )$ for all $\mathbf {c}=\{c_k\}\in \ell ^{p}$, and $\Vert f\Vert _{p,(n+1)(\theta _1-1)}\preceq \Vert \mathbf {c}\Vert _{p}$.

Since, by assumption, $T^\beta _\mu$ is bounded from $A^{p}\bigl (D,(n+1)(\theta _1-1)\bigr )$ to $A^{q}\bigl (D,(n+1)(\theta _2-1)\bigr )$ we have

$$\begin{aligned}\begin{aligned} \left\| T^\beta _\mu f_t\right\| ^{q}_{q,(n+1)(\theta _2-1)}&=\int _D\left| \sum _{k=0}^\infty c_k r_k(t) T^\beta _\mu f_{a_k}(z)\right| ^{q}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z)\\&\le \Vert T^\beta _\mu \Vert ^{q}\Vert f_t\Vert ^{q}_{p,(n+1)(\theta _1-1)}\preceq \Vert T^\beta _\mu \Vert ^{q}\Vert \mathbf {c}\Vert ^{q}_{p}. \end{aligned} \end{aligned}$$

Integrating both sides on [0, 1] with respect to t and using Khinchine’s inequality (see, e.g., [18]) we obtain

$$\begin{aligned} \int _D\left( \sum _{k=0}^\infty |c_k|^2|T^\beta _\mu f_{a_k}(z)|^2\right) ^{q/2}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z)\preceq \Vert T^\beta _\mu \Vert ^{q}\Vert \mathbf {c}\Vert ^{q}_{p}. \end{aligned}$$

Set $B_k=B_D(a_k,r)$. We have to consider two cases: $q\ge 2$ and $0<q<2$.

If $q\ge 2$, using the fact that $\Vert \mathbf {a}\Vert _{q/2}\le \Vert \mathbf {a}\Vert _1$ for every $\mathbf {a}\in \ell ^1$ we get

$$\begin{aligned}\begin{aligned} \sum _{k=0}^\infty |c_k|^{q}\int _{B_k}|&T^\beta _\mu f_{a_k}(z)|^{q}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z)\\&\le \int _D\left( \sum _{k=0}^\infty |c_k|^2|T^\beta _\mu f_{a_k}(z)|^2 \chi _{B_k}(z)\right) ^{q/2}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z)\\&\le \int _D\left( \sum _{k=0}^\infty |c_k|^2|T^\beta _\mu f_{a_k}(z)|^2\right) ^{q/2}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z). \end{aligned}\end{aligned}$$

If instead $0<q<2$, using Hölder’s inequality, we obtain

$$\begin{aligned} \begin{aligned} \sum _{k=0}^\infty |c_k|^{q}\int _{B_k}&|T^\beta _\mu f_{a_k}(z)|^{q}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z)\\&\le \int _D\left( \sum _{k=0}^\infty |c_k|^2|T^\beta _\mu f_{a_k}(z)|^2\right) ^{\frac{q}{2}}\left( \sum _{k=0}^\infty \chi _{B_k}(z)\right) ^{1-\frac{q}{2}}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z)\\&\preceq \int _D\left( \sum _{k=0}^\infty |c_k|^2|T^\beta _\mu f_{a_k}(z)|^2\right) ^{q/2}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z), \end{aligned} \end{aligned}$$

where we used the fact that each $z\in D$ belongs to no more than m of the $B_k$.

Summing up, for any $q>0$ we have

$$\begin{aligned} \sum _{k=0}^\infty |c_k|^{q}\int _{B_k}|T^\beta _\mu f_{a_k}(z)|^{q}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z)\preceq \Vert T^\beta _\mu \Vert ^{q}\Vert \mathbf {c}\Vert ^{q}_{p}. \end{aligned}$$

Now Lemmas 2.1, 2.2 and 2.5 (see also [4, Corollary 2.7]) yield

$$\begin{aligned} |T^\beta _\mu f_{a_k}(a_k)|^{q}\preceq \delta (a_k)^{-(n+1)\theta _2} \int _{B_k} |T^\beta _\mu f_{a_k}(z)|^{q}\delta (z)^{(n+1)(\theta _2-1)}\,d\nu (z), \end{aligned}$$

and so we get

$$\begin{aligned} \sum _{k=0}^\infty |c_k|^{q}\delta (a_k)^{(n+1)\theta _2}|T^\beta _\mu f_{a_k}(a_k)|^{q}\preceq \Vert T^\beta _\mu \Vert ^{q}\Vert \mathbf {c}\Vert ^{q}_{p}. \end{aligned}$$

On the other hand, using Lemmas 2.6 and 2.7, we obtain

$$\begin{aligned}\begin{aligned} T^\beta _\mu f_{a_k}(a_k)&=\delta (a_k)^\tau \int _D K(a_k,w)^\beta k_{a_k}(w)^\beta \,d\mu (w)\\&\succeq \delta (a_k)^{\tau +(n+1)\frac{\beta }{2}} \int _D |K(a_k,w)|^{2\beta } \,d\mu (w)\\&\ge \delta (a_k)^{\tau +(n+1)\frac{\beta }{2}}\int _{B_D(a_k,r)} |K(a_k,w)|^{2\beta }\,d\mu (w)\\&\succeq \delta (a_k)^{\tau -(n+1)\frac{3\beta }{2}}\mu \bigl (B_D(a_k,r)\bigr )=\delta (a_k)^{-(n+1)[\beta +\frac{\theta _1}{p}]}\mu \bigl (B_D(a_k,r)\bigr ). \end{aligned}\end{aligned}$$

Putting all together we get

$$\begin{aligned} \sum _{k=0}^\infty |c_k|^{q}\left( \frac{\mu \bigl (B_D(a_k,r)\bigr )}{\delta (a_k)^{(n+1)\lambda \gamma }}\right) ^{q}\preceq \Vert T^\beta _\mu \Vert ^{q}\Vert \mathbf {c}\Vert ^{q}_{p}. \end{aligned}$$

Set $\mathbf {d}=\{d_k\}$, where

$$\begin{aligned} d_k=\frac{\mu \bigl (B_D(a_k,r)\bigr )}{\delta (a_k)^{(n+1)\lambda \gamma }}. \end{aligned}$$

Then by duality we get $\{d_k^{q}\}\in \ell ^{p/(p-q)}$ with $\Vert \{d_k^{q}\}\Vert _{p/(p-q)}\preceq \Vert T^\beta _\mu \Vert ^{q}$, because $p/(p-q)$ is the conjugate exponent of $p/q>1$. This means that $\mathbf {d}\in \ell ^{pq/(p-q)}=\ell ^{1/(1-\lambda )}$ with

$$\begin{aligned} \Vert \mathbf {d}\Vert _{\frac{1}{1-\lambda }}\preceq \Vert T^\beta _\mu \Vert . \end{aligned}$$

Since

$$\begin{aligned} d_k \approx {\hat{\mu }}_r(a_k)\delta (a_k)^{(n+1)(1-\lambda \gamma )}, \end{aligned}$$

the assertion then follows from Theorem 2.16. $\square$

Remark 3.5

Note that a similar result holds also for $\lambda \ge 1$ 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 $S^{s,r}_{\mu ,t}$ that will be crucial for the proof of our main theorem.

Proposition 3.6

Let $D\subset \subset \mathbb {C}^n$ be a bounded strongly pseudoconvex domain, and $\mu$ a positive finite Borel measure on D. Fix $q>1$, $p>0$, $\theta _1$, $\theta _2>1-\frac{1}{n+1}$ and $r,s>0$, and $t>\frac{1}{p}\max \left\{ 1,\theta _2,p-1+\theta _2\right\} >0$. Set

$$\begin{aligned} \lambda =1+\frac{r}{p}-\frac{1}{q} \quad \mathrm {and}\quad \gamma =\frac{1}{\lambda } \left( \frac{t}{2}+\frac{\theta _2 r}{p}-\frac{\theta _1}{q}-s\right) . \end{aligned}$$

Assume that $\lambda >0$ and $\gamma >1-\frac{1}{n+1}$, and that there exists $K>0$ such that

$$\begin{aligned} \Big \Vert S^{s,r}_{\mu ,t}f\Big \Vert _{q,(n+1)(\theta _1-1)}\le K\Vert f\Vert ^r_{p,(n+1)(\theta _2-1)} \end{aligned}$$

(3.5)

for all $f\in A^p\bigl (D, (n+1)(\theta _2-1)\bigr )$. Then $\mu$ is a $(\lambda ,\gamma )$-skew Carleson measure with $\Vert \mu \Vert _{\lambda ,\mu }\preceq K$.

Proof

Let us first consider the case $\lambda \ge 1$. Given $a\in D$ and $\sigma \in \mathbb {N}$ such that

$$\begin{aligned} p\sigma >\theta _2, \end{aligned}$$

set

$$\begin{aligned} f_a^\sigma (z)=k_a(z)^\sigma , \end{aligned}$$

for $z\in D$. By Proposition 2.9 we have

$$\begin{aligned} \Vert f_a^\sigma \Vert ^r_{p,(n+1)(\theta _2-1)}\preceq \delta (a)^{(n+1)(\theta _2\frac{r}{p}-\frac{r\sigma }{2})}. \end{aligned}$$

(3.6)

Now fix $\rho >0$. Clearly, there is a $\hat{\rho }>0$ depending only on $\rho$ such that z, $w\in B_D(a,\rho )$ implies $w\in B_D(z,\hat{\rho })$ for all $a\in D$. By Lemma 2.2 we can find $\delta _1>0$ such that if $\delta (a)<\delta _1$ then $\delta (z)<\delta _{\hat{\rho }}$ for all $z\in B_D(a,\rho )$, where $\delta _{\hat{\rho }}>0$ is given by Lemma 2.7. Then if $\delta (a)<\delta _1$ using Lemmas 2.1, 2.2 and 2.7 we have

$$\begin{aligned} \Big \Vert S^{s,r}_{\mu ,t}f^\sigma _a\Big \Vert ^q_{q,(n+1)(\theta _1-1)}&=\int _D \Big |S^{s,r}_{\mu ,t}f^\sigma _a(z)\Big |^q\delta (z)^{(n+1)(\theta _1-1)}\, d\nu (z)\\&\ge \int _{B_D(a,\rho )} \Big |S^{s,r}_{\mu ,t}f^\sigma _a(z)\Big |^q \delta (z)^{(n+1)(\theta _1-1)}\, d\nu (z)\\&\succeq \delta (a)^{(n+1)(\theta _1-1)}\int _{B_D(a,\rho )} \delta (z)^{(n+1)qs}\\&\quad \left[ \int _D |k_z(w)|^t|f_a^\sigma (w)|^r\,d\mu (w)\right] ^q d\nu (z)\\&\succeq \delta (a)^{(n+1)(\theta _1-1+qs)} \int _{B_D(a,\rho )} \left[ \int _{B_D(a,\rho )} |k_z(w)|^t|f_a^\sigma (w)|^r\,d\mu (w)\right] ^q d\nu (z)\\&\succeq \delta (a)^{(n+1)(\theta _1-1+qs -\frac{1}{2}\sigma rq)}\int _{B_D(a,\rho )}\delta (z)^{\frac{n+1}{2}tq}\\&\quad \left[ \int _{B_D(a,\rho )}|K(z,w)|^t\,d\mu (w)\right] ^q d\nu (z)\\&\succeq \delta (a)^{(n+1)(\theta _1-1+qs -\frac{1}{2}\sigma rq+\frac{1}{2}tq)} \int _{B_D(a,\rho )}\delta (z)^{-(n+1)tq}\mu \bigl (B_D(a,\rho )\bigr )^q\,d\nu (z)\\&\succeq \delta (a)^{(n+1)(\theta _1+qs-\frac{1}{2}\sigma rq-\frac{1}{2}tq)}\mu \bigl (B_D(a,\rho )\bigr )^q. \end{aligned}$$

Recalling (3.5) and (3.6), we get

$$\begin{aligned} \begin{aligned} \mu \bigl (B_D(a,\rho )\bigr )&\preceq K\delta (a)^{(n+1)(\frac{\sigma r}{2}+\frac{t}{2}-s-\frac{\theta _1}{q})}\Vert f_a^\sigma \Vert _{p,(n+1)(\theta _2-1)}^r\\&\preceq K\delta (a)^{(n+1)(\frac{\sigma r}{2}+\frac{t}{2}-s-\frac{\theta _1}{q} +\theta _2\frac{r}{p}-\frac{r\sigma }{2})}\\&\preceq K\nu \bigl (B_D(a,\rho )\bigr )^{\frac{t}{2}-s-\frac{\theta _1}{q} +\theta _2\frac{r}{p}}. \end{aligned} \end{aligned}$$

Since $\mu$ is a finite measure, a similar estimate holds when $\delta (a)\ge \delta _1$. Then Theorem 2.15 implies that $\mu$ is $(\lambda ,\gamma )$-skew Carleson with $\Vert \mu \Vert _{\lambda ,\gamma }\preceq K$ as claimed.

Now let us assume $0<\lambda <1$. Assume first $r=1$. Choose $\beta \in {\mathbb {N}}$ with $\beta \ge t$ and set

$$\begin{aligned} \sigma =\theta _1+q\left( \beta -\frac{t}{2}+s\right) >1-\frac{1}{n+1}. \end{aligned}$$

We can apply (3.5) and Lemma 3.2.(i) with $p=q$ to get

$$\begin{aligned} \left\| T^\beta _\mu f\right\| _{q,(n+1)(\sigma -1)}\preceq K \Vert f\Vert _{p,(n+1)(\theta _2-1)}. \end{aligned}$$

Therefore Proposition 3.4 implies that $\mu$ is $(\lambda ,{\tilde{\gamma }})$-skew Carleson with

$$\begin{aligned} {\tilde{\gamma }}=\frac{1}{\lambda }\left( \beta +\frac{\theta _2}{p}-\frac{\sigma }{q}\right) =\frac{1}{\lambda }\left( \frac{t}{2}+\frac{\theta _2}{p}-\frac{\theta _1}{q}-s\right) =\gamma , \end{aligned}$$

and $\Vert \mu \Vert _{\lambda ,\gamma }\preceq K$ as claimed.

Assume now $r>1$, and choose $\alpha >0$ so that

$$\begin{aligned} \beta =\frac{t}{r}+\frac{\alpha }{r'}>\frac{1}{p}\max \left\{ 1,\theta _2,p-1+\theta _2\right\} \end{aligned}$$

and $\beta \in \mathbb {N}$. We also require that $\alpha$ is such that $\alpha >\lambda \gamma +\frac{n}{n+1}(1-\lambda )$ and

$$\begin{aligned} \sigma :=\tau \left[ \frac{\alpha -\lambda \gamma }{r'}+\frac{1}{qr}\left( \theta _1+q\left( s+\frac{t}{2}\right) \right) \right] >1-\frac{1}{n+1}, \end{aligned}$$

where

$$\begin{aligned} \tau =\frac{1}{1-\lambda +\frac{1}{p}}. \end{aligned}$$

Assume for a moment that $\mu$ has compact support. Then $\Vert \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )}B^\alpha \mu \Vert _{\frac{1}{1-\lambda },(n+1)(\gamma -1)}$ is finite; therefore (3.5) and Corollary 3.3 applied with ${\tilde{p}}=p$ imply that $T^\beta _\mu$ is bounded from $A^p\bigl (D,(n+1)(\theta _2-1)\bigr )$ to $A^\tau \bigl (D,(n+1)(\sigma -1)\bigr )$, with

$$\begin{aligned} \Vert T^\beta _\mu \Vert \preceq K\left\| \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )}B^\alpha \mu \right\| _{\frac{1}{1-\lambda },(n+1)(\gamma -1)}^{1/r'}. \end{aligned}$$

Proposition 3.4 then yields that $\mu$ is $({\tilde{\lambda }},{\tilde{\gamma }})$-skew Carleson with

$$\begin{aligned} {\tilde{\lambda }}=1+\frac{1}{p}-\frac{1}{\tau }=\lambda \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\tilde{\gamma }}&=\frac{1}{\lambda }\left( \beta +\frac{\theta _2}{p}-\frac{\sigma }{\tau }\right) =\frac{1}{\lambda }\left( \beta +\frac{\theta _2}{p}-\frac{\alpha }{r'}+\frac{\lambda \gamma }{r'}- \frac{\theta _1}{qr}-\frac{1}{r}\left( s+\frac{t}{2}\right) \right) \\&=\frac{1}{\lambda }\left( \frac{t}{2r}+\frac{\theta _2}{p}+\frac{t}{2r'}+\frac{\theta _2 r}{pr'}-\frac{\theta _1}{qr'}-\frac{s}{r'}-\frac{\theta _1}{qr}-\frac{s}{r}\right) \\&=\frac{1}{\lambda }\left( \frac{t}{2}-\frac{\theta _1}{q}-s+\frac{\theta _2 r}{p}\right) =\gamma . \end{aligned} \end{aligned}$$

Furthermore, we also have

$$\begin{aligned} \left\| \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )}B^\alpha \mu \right\| _{\frac{1}{1-\lambda },(n+1)(\gamma -1)}\preceq K \left\| \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )}B^\alpha \mu \right\| _{\frac{1}{1-\lambda },(n+1)(\gamma -1)}^{1/r'} \end{aligned}$$

and thus

$$\begin{aligned} \left\| \delta ^{-(n+1)\big (\gamma -\frac{\alpha }{2}\big )}B^\alpha \mu \right\| _{\frac{1}{1-\lambda },(n+1)(\gamma -1)}\preceq K. \end{aligned}$$

An easy limit argument then shows that this holds even when the support of $\mu$ is not compact, and then, by Theorem 2.16, $\mu$ is $(\lambda ,\gamma )$-skew Carleson with $\Vert \mu \Vert _{\lambda ,\gamma }\preceq K$.

We are left with the case $0<r<1$. Choose $R>1$ and set $\mu ^*=\delta ^A\mu$, with

$$\begin{aligned} A=(n+1)\frac{(R-r)\theta _2}{p}. \end{aligned}$$

First of all, fix $r_0\in (0,1)$ and set $R_0=\frac{1}{2}(1+r_0)$. Then, for any $z\in D$, Lemmas 2.1, 2.2 and 2.5 yield

$$\begin{aligned} \begin{aligned} |f(z)|^p&\preceq \frac{1}{\nu \bigl (B_D(z,r_0)\bigr )}\int _{B_D(z,R_0)}|f(w)|^p\,d\nu (w)\\&\preceq \delta (z)^{-(n+1)\theta _2}\int _{B_D(z,R_0)}|f(w)|^p \delta (w)^{(n+1)(\theta _2-1)}\,d\nu (w)\\&\le \delta (z)^{-(n+1)\theta _2}\Vert f\Vert ^p_{p,(n+1)(\theta _2-1)}. \end{aligned} \end{aligned}$$

Then (3.5) yields

$$\begin{aligned} {\begin{aligned} \Big \Vert S^{s,R}_{\mu ^*,t}&f\Big \Vert _{q,(n+1)(\theta _1-1)}^q\\&=\int _D\left( \delta (z)^{(n+1)s}\int _D |k_z(w)|^t |f(w)|^r |f(w)|^{R-r}\,d\mu ^*(w) \right) ^q \delta (z)^{(n+1)(\theta _1-1)}\,d\nu (z)\\&\preceq \Vert f\Vert ^{q(R-r)}_{p,(n+1)(\theta _2-1)}\int _D\!\!\!\left( \!\delta (z)^{(n+1)s}\!\!\int _D\!\! |k_z(w)|^t |f(w)|^r \delta (w)^{A-(n+1)\frac{(R-r)\theta _2}{p}}d\mu (w)\!\right) ^q\\&\quad \delta (z)^{(n+1)(\theta _1-1)}d\nu (z)\\&=\Vert f\Vert ^{q(R-r)}_{p,(n+1)(\theta _2-1)}\Big \Vert S^{s,r}_{\mu ,t}f\Big \Vert ^q_{q,(n+1)(\theta _1-1)}\\&\le K \Vert f\Vert ^{qR}_{p,(n+1)(\theta _2-1)} \end{aligned}}\end{aligned}$$

for all $f\in A^p\bigl (D,(n+1)(\theta _2-1)\bigr )$. Arguing as before, we can prove that $\mu ^*$ is $(\lambda ,\gamma ^*)$-skew Carleson with $\Vert \mu ^*\Vert _{\lambda ,\gamma ^*}\preceq K$, where

$$\begin{aligned} \gamma ^*=\frac{1}{\lambda } \left( \frac{t}{2}+\frac{\theta _2R}{p}-\frac{\theta _1}{q}-s\right) =\gamma +\frac{(R-r)\theta _2}{\lambda p}. \end{aligned}$$

But $\mu =\delta ^{-(n+1)\frac{(R-r)\theta _2}{p}}\mu ^*$; then Lemma 2.18 implies that $\mu$ is $(\lambda ,\gamma )$-skew Carleson with $\Vert \mu \Vert _{\lambda ,\gamma }\preceq K$, and we are done. $\square$

We finally have all the ingredients to prove our main result.

Proof of Theorem 1.1

Assume that $\mu$ is $(\lambda ,\gamma )$-skew Carleson. For $k=1$ the assertion is just the definition of $(\lambda ,\gamma )$-skew Carleson; so we can assume $k\ge 2$.

For $j=1,\ldots ,k$ put $\beta _j=\lambda \frac{p_j}{q_j}$. Then we have $\beta _j>1$, $\frac{q_j}{p_j}\beta _j=\lambda$, and

$$\begin{aligned} \sum _{j=1}^k \frac{1}{\beta _j}=1. \end{aligned}$$

Now define $\eta _j\in \mathbb {R}$ as

$$\begin{aligned} \eta _j=\frac{q_j}{p_j}\theta _j-\frac{1}{\beta _j}\lambda \gamma = \frac{q_j}{p_j}(\theta _j-\gamma ); \end{aligned}$$

in particular

$$\begin{aligned} \gamma +\frac{1}{\lambda }\beta _j\eta _j=\theta _j. \end{aligned}$$

(3.7)

It is easy to check that $\eta _1+\cdots +\eta _k=0$; then Hölder’s inequality yields

$$\begin{aligned} \int _D \prod _{j=1}^k |f_j(z)|^{q_j}\,d\mu (z)\le \prod _{j=1}^k\left[ \int _D |f_j(z)|^{\beta _jq_j}\delta (z)^{\beta _j\eta _j}\,d\mu (z)\right] ^{1/\beta _j}. \end{aligned}$$

(3.8)

Now, Lemma 2.18 implies that $\delta ^{\beta _j\eta _j}\mu$ is $(\lambda ,\gamma +\frac{1}{\lambda }\beta _j\eta _j)$-skew Carleson, that is, $(\lambda ,\theta _j)$-skew Carleson, by (3.7). But $\lambda =\frac{q_j\beta _j}{p_j}$; hence Theorems 2.15 and 2.16 imply that $\delta ^{\beta _j\eta _j}\mu$ is $(p_j,q_j\beta _j;\alpha _j)$-skew Carleson, with $\alpha _j=(n+1)(\theta _j-1)$. Therefore

$$\begin{aligned} \left[ \int _D |f_j(z)|^{\beta _j q_j}\delta (z)^{\beta _j\eta _j}\,d\mu (z)\right] ^{1/\beta _j}\preceq \Vert f_j\Vert ^{q_j}_{p_j,(n+1)(\theta _j-1)} \end{aligned}$$

(3.9)

for $j=1,\ldots , k$, and (1.1) is proved (see also Remark 3.7 below).

Assume now that (1.1) holds for any $f_j\in A^{p_j}\bigl (D,(n+1)(\theta _j-1)\bigr )$ with $j=1,\ldots ,k$; we would like to prove by induction that $\mu$ is a $(\lambda ,\gamma )$-skew Carleson measure with $\Vert \mu \Vert _{\lambda ,\gamma }\preceq C$. If $k=1$ there is nothing to prove, so we can assume $k\ge 2$.

Assume first $\lambda \ge 1$, and let $\alpha _j=(n+1)(\theta _j-1)$ for $j=1,\ldots ,k$. Choose $\sigma _1,\ldots ,\sigma _k\in \mathbb {N}^*$ such that

$$\begin{aligned} p_j\sigma _j>\max \{1,\theta _j\} \end{aligned}$$

for all $j=1,\ldots ,k$, and

$$\begin{aligned} \sum _{j=1}^k q_j\sigma _j>\lambda \gamma , \end{aligned}$$

and set

$$\begin{aligned} r_j=(n+1)\left[ \frac{\sigma _j}{2}-\frac{\theta _j}{p_j}\right] \end{aligned}$$

for all $j=1,\ldots ,k$.

For any $a\in D$ and $j=1,\ldots ,k$, consider

$$\begin{aligned} f_{j,a}(z)=k_a(z)^{\sigma _j}\delta (a)^{r_j}. \end{aligned}$$

Then, since $\alpha _j<(n+1)(p_j\sigma _j-1)$ by the choice of $\sigma _j$, applying Proposition 2.9 we obtain

$$\begin{aligned} \Vert k_a^{\sigma _j}\Vert _{p_j,\alpha _j}=\Vert k_a\Vert ^{\sigma _j}_{p_j\sigma _j,\alpha _j}\preceq \delta (a)^{\frac{1}{p_j}\left[ \alpha _j-(n+1)\left( \frac{p_j\sigma _j}{2}-1\right) \right] } =\delta (a)^{-r_j}, \end{aligned}$$

and hence

$$\begin{aligned} \Vert f_{j,a}\Vert _{p_j,\alpha _j}\preceq 1 \end{aligned}$$

for $j=1,\ldots ,k$. Thus (1.1) yields

$$\begin{aligned} \int _D\prod _{j=1}^k |f_{j,a}(z)|^{q_j}\,d\mu (z)\le C \prod _{j=1}^k\Vert f_{j,a}\Vert ^{q_j}_{p_j,\alpha _j} \preceq C. \end{aligned}$$

(3.10)

Now recall that

$$\begin{aligned} \prod _{j=1}^k |f_{j,a}(z)|^{q_j}=|k_a(z)|^{\sum _j q_j\sigma _j}\delta (a)^{\sum _j q_jr_j}. \end{aligned}$$

We have

$$\begin{aligned} \sum _{j=1}^k q_jr_j=(n+1)\sum _{j=1}^k\left[ \frac{q_j\sigma _j}{2}-\theta _j\frac{q_j}{p_j}\right] =\frac{n+1}{2}\sum _{j=1}^k q_j\sigma _j-(n+1)\lambda \gamma , \end{aligned}$$

so, setting $s=\sum _j \sigma _jq_j$, (3.10) becomes

$$\begin{aligned} \delta ^{(n+1)\left( \frac{s}{2}-\lambda \gamma \right) }B^s\mu \preceq C, \end{aligned}$$

and Theorem 2.15 implies that $\mu$ is $(\lambda ,\gamma )$-Carleson with

$$\begin{aligned} \Vert \mu \Vert _{\lambda ,\gamma } \approx \Vert \delta ^{(n+1)\left( \frac{s}{2}-\lambda \gamma \right) }B^s\mu \Vert _\infty \preceq C. \end{aligned}$$

We are left with the case $0<\lambda <1$. We argue again by induction on k. If $k=1$, it is the definition of skew Carleson measure; so assume the assertion holds for $k-1$. Set

$$\begin{aligned} {\tilde{\lambda }}=\sum _{j=1}^{k-1}\frac{q_j}{p_j}\quad \mathrm {and}\quad {\tilde{\gamma }}=\frac{1}{{\tilde{\lambda }}}\sum _{j=1}^{k-1}\theta _j\frac{q_j}{p_j}. \end{aligned}$$

Fix a function $g\in A^{p_k}\bigl (D,(n+1)(\theta _k-1)\bigr )$, and set $\mu _k=|g|^{q_k}\mu$. Then (1.1) yields

$$\begin{aligned} \int _D \prod _{j=1}^{k-1}|f_j(z)|^{q_j}\,d\mu _k(z)\le C \Vert g\Vert ^{q_k}_{p_k,(n+1)(\theta _k-1)} \prod _{j=1}^{k-1}\Vert f_j\Vert ^{q_j}_{p_j,(n+1)(\theta _j-1)} \end{aligned}$$

for all $f_j\in A^{p_j}\bigl (D,(n+1)(\theta _j-1)\bigr )$ with $j=1,\ldots ,k-1$. By induction, this means that $\mu _k$ is a $({\tilde{\lambda }},{\tilde{\gamma }})$-skew Carleson measure with $\Vert \mu _k\Vert _{{\tilde{\lambda }},{\tilde{\gamma }}}\preceq C\Vert g\Vert ^{q_k}_{p_k,(n+1)(\theta _k-1)}$. Since ${\tilde{\lambda }}<\lambda <1$, and ${\tilde{\gamma }}>1-\frac{1}{n+1}$, Theorem 2.16 implies that $\delta ^{-(n+1)({\tilde{\gamma }}-\frac{t}{2})}B^t\mu _k\in L^{1/(1-{\tilde{\lambda }})} \bigl (D, (n+1)({\tilde{\gamma }}-1)\bigr )$ for all $t>{\tilde{\lambda }}{\tilde{\gamma }}+\frac{n}{n+1}(1-{\tilde{\lambda }})$, with

$$\begin{aligned} \left\| \delta ^{-(n+1)({\tilde{\gamma }}-\frac{t}{2})}B^t\mu _k\right\| _{1/(1-{\tilde{\lambda }}), (n+1)({\tilde{\gamma }}-1)}\preceq C\Vert g\Vert ^{q_k}_{p_k,(n+1)(\theta _k-1)}. \end{aligned}$$

Writing explicitely the previous formula we obtain

$$\begin{aligned}&\left[ \int _D\left[ \int _D |k_a(z)|^t|g(z)|^{q_k}\,d\mu (z)\right] ^{1/(1-{\tilde{\lambda }})} \delta (a)^{-\frac{n+1}{1-{\tilde{\lambda }}}({\tilde{\gamma }}-\frac{t}{2})} \delta (a)^{(n+1)({\tilde{\gamma }}-1)} \,d\nu (a)\right] ^{1-{\tilde{\lambda }}} \\&\quad \preceq C\Vert g\Vert ^{q_k}_{p_k,(n+1)(\theta _k-1)}, \end{aligned}$$

that is

$$\begin{aligned} \Vert S^{s,q_k}_{\mu ,t}g\Vert _{1/(1-{\tilde{\lambda }}),(n+1)({\tilde{\gamma }}-1)} \preceq C\Vert g\Vert ^{q_k}_{p_k,(n+1)(\theta _k-1)}, \end{aligned}$$

where $s=\frac{t}{2}-{\tilde{\gamma }}$. Choosing $t>\frac{1}{p_k}\max \{1,\theta _k,p_k-1-\theta _k\}$ such that $s>0$, we deduce from Proposition 3.6 that $\mu$ is a $(\lambda ^*,\gamma ^*)$-skew Carleson measure with $\Vert \mu \Vert _{\lambda ^*,\gamma ^*}\preceq C$, where

$$\begin{aligned} \lambda ^*=1+\frac{q_k}{p_k}-(1-{\tilde{\lambda }})=\lambda \quad \mathrm {and}\quad \gamma ^*=\frac{1}{\lambda ^*}\left( \theta _k\frac{q_k}{p_k}+{\tilde{\gamma }}{\tilde{\lambda }}\right) =\gamma , \end{aligned}$$

and we are done. $\square$

Remark 3.7

If $\mu$ is a $(\lambda ,\gamma )$-skew Carleson measure, we can estimate the constant C in (1.1). Fix $r\in (0,1)$. Then Lemmas 2.1 and 2.2 yield

$$\begin{aligned} (\widehat{\delta ^{\beta _j\eta _j}\mu })_r \approx \delta ^{(n+1)\beta _j\eta _j}{\hat{\mu }}_r. \end{aligned}$$

If $\lambda \ge 1$ we can now use (2.2) to get

$$\begin{aligned} \begin{aligned} \Vert \delta ^{\beta _j\eta _j}\mu \Vert _{p_j,q_j\beta _j;\alpha _j}&\approx \left\| \delta ^{-(n+1)(\lambda \theta _j-1)}(\widehat{\delta ^{\beta _j\eta _j}\mu })_r \right\| _\infty \\&\approx \left\| \delta ^{-(n+1)(\lambda \theta _j-1-\beta _j\eta _j)}{\hat{\mu }}_r\right\| _\infty = \Vert \delta ^{-(n+1)(\lambda \gamma -1)}{\hat{\mu }}_r\Vert _\infty \approx \Vert \mu \Vert _{\lambda ,\gamma }. \end{aligned} \end{aligned}$$

Analogously, if $0<\lambda <1$ we can use (2.3) to get

$$\begin{aligned} {\begin{aligned} \Vert \delta ^{\beta _j\eta _j}\mu \Vert _{p_j,q_j\beta _j;\alpha _j}&\approx \left\| \delta ^{-(n+1)(\theta _j-1)\lambda }(\widehat{\delta ^{\beta _j\eta _j}\mu })_r \right\| _{\frac{1}{1-\lambda }}\\&\approx \left\| \delta ^{-(n+1)(\lambda \theta _j-\lambda -\beta _j\eta _j)}{\hat{\mu }}_r\right\| _{\frac{1}{1-\lambda }} = \left\| \delta ^{-(n+1)(\gamma -1)\lambda }{\hat{\mu }}_r\right\| _{\frac{1}{1-\lambda }} \approx \Vert \mu \Vert _{\lambda ,\gamma }. \end{aligned}}\end{aligned}$$

Therefore in both cases (3.8) and (3.9) yield

$$\begin{aligned} C\approx \Vert \mu \Vert _{\lambda ,\gamma }^{\sum _j q_j}. \end{aligned}$$