The Dominating Set of Bergman Spaces

Xin Song

Hebei University of Technology, Tianjin, China.

DOI: 10.4236/jamp.2019.77106                

Abstract

The dominating set of the weighted Bergman space in the unit disk is characterized in terms of the pseudo-hyperbolic metric disk. Our method is to generalize Luecking’s three key lemmas on Bergman space to the weighted Bergman space in the unit disk. We then apply those three lemmas to give a complete description of the dominating set of the weighted Bergman space.

Keywords

Bergman Space, Carleson Measure, Reverse Carleson Inequality

1. Introduction

Let $\mathbb{D}$ be the open unit disk in the complex plane $ℂ$ and let $\text{d}A$ be the Lebesgue area measure on $\mathbb{D}$ . For $\alpha >-1$ the weighted Lebesgue measure $\text{d}{A}_{\alpha }$ is defined by

$\text{d}{A}_{\alpha }=c_{\alpha }{\left(1-{\left|z\right|}^2\right)}^{\alpha }\text{d}A\left(z\right)\mathrm{,}$

where

$c_{\alpha }=\frac{\Gamma \left(2+\alpha \right)}{\Gamma \left(\alpha +1\right)}$

is a normalizing constant so that

$A_{\alpha }\left(\mathbb{D}\right)={\displaystyle {\int }_{\mathbb{D}}}\text{d}{A}_{\alpha }=1.$

If $\mu$ is a positive measure on $\mathbb{D}$ and $p>0$ , we denote $L^p\left(\mu \right)$ the Lebesgue space over $\mathbb{D}$ with respect to $\mu$ . That is, $L^p\left(\mu \right)$ consists of all functions f defined on $\mathbb{D}$ for which

${\Vert f\Vert }_{L^p\left(\mu \right)}={\left({\displaystyle {\int }_{\mathbb{D}}}{\left|f\left(z\right)\right|}^p\text{d}\mu \left(z\right)\right)}^{1/p}<\infty \mathrm{.}$

For $\alpha >-1$ and $p>0$ , the weighted Bergman space ${\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)$ is defined by ${\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)=H\left(\mathbb{D}\right)\cap L^p\left(\mathbb{D}\mathrm{,}\text{d}{A}_{\alpha }\right)$ , where $H\left(\mathbb{D}\right)$ is the space of all analytic functions on $\mathbb{D}$ . That is $f\in {\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)$ if it is holomorphic and

${\Vert f\Vert }_{{\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)}\mathrm{<mo>\; :\; </mo>}={\left({\displaystyle {\int }_{\mathbb{D}}}{\left|f\left(z\right)\right|}^p\text{d}{A}_{\alpha }\left(z\right)\right)}^{\frac{1}{p}}<\infty \mathrm{.}$

For any $a\in \mathbb{D}$ and $r\in \left(\mathrm{0,1}\right)$ we write

$\Delta \left(a\mathrm{,}r\right)=\left\{z\in \mathbb{D}\mathrm{<mo>\; :\; </mo>}\rho \left(z\mathrm{,}a\right)<r\right\}\mathrm{,}$

where $\rho \left(z\mathrm{,}a\right)=\left|{\phi }_a\left(z\right)\right|$ is the pseudo-hyperbolic metric and ${\phi }_a\left(z\right)=\left(a-z\right)/\left(1-\bar{a}z\right)$ .

Let $I\mathrm{<mo>\; :\; </mo>}{\mathcal{A}}^p\left(\mathbb{D}\right)\to L^q\left(\text{d}\mu \right)$ be an identity, we say $\mu$ is a ${\mathcal{A}}^p\left(\mathbb{D}\right)$ -Carleson measure, if there is a constant $C>0$ such that

${\Vert f\Vert }_{{\mathcal{A}}^p\left(\mathbb{D}\right)}\le C{\Vert If\Vert }_{L^q(d\mu )}$

for each $f\in {\mathcal{A}}^p\left(\mathbb{D}\right)$ .

Now we define the dominanting set of Bergman spaces.

Lemma 1 Let $p>0$ , $\alpha >0$ and G be a Lebesgue measurable subset of $\mathbb{D}$ . We call G is a dominanting set of ${\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)$ if there is a constant $C>0$ such that

${\displaystyle {\int }_{\mathbb{D}}}{\left|f\right|}^p\text{d}{A}_{\alpha }\le C{\displaystyle {\int }_G}{\left|f\right|}^p\text{d}{A}_{\alpha }$ (1)

for all $f\in {\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)$ .

Let ${\chi }_G$ is the characteristic function of G. According to the definition of dominanting set, measure $\text{d}\mu ={\chi }_G\text{d}v$ satisfies the reverse inequality in Carleson measure definition, that is we have

${\displaystyle {\int }_{\mathbb{D}}}{\left|f\right|}^p\text{d}{A}_{\alpha }\le C{\displaystyle {\int }_{\mathbb{D}}}{\left|f\right|}^p\text{d}\mu$

for all $f\in {\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)$ . We call reverse Carleson measure if the measure satidfy reverse Carleson inequality. The purpose of this paper is to study reverse ${\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)$ -Carleson measure. [1] and [2] provide some basic tools of Bergman space and some analytic functions. [3] proved Carleson-type embedding theorems for weighted Bergman spaces with Bkoll weights. In 1985, Luecking [4] first studied the reverse Carleson measure in Bergman space on unit disk. The main research tool of Luecking is the dominanting set in Bergman space. In [5] , Luecking introduces the necessary and sufficient conditions for dominating sets in Bergman spaces on a unit disk. In recent years, Lou and Zhuo [6] generalized this work to Fock space and gave the characterization of dominant set in Fock space. The purpose of this paper is to extend dominating set and reverse Carleson measure to weighted Bergman spaces. We can find some other concepts in [7] [8] [9] [10] [11] . [12] is a survey on reverse Carleson measures for various Hilbert spaces of analytic functions. We can use some definitions and proof methods in the paper to prove our results. [13] discussed direct and reverse Carleson measures for the de Branges-Rovnyak spaces $\mathcal{H}\left(b\right)$ . We can refer to their method. In [14] , Korhonen and Rättyä has proved the sampling measure by using dominating set and p-Carleson measure for weighted Bergman space with a weight $\omega$ , reference resources [15] [16] [17] [18] .

The main results is as follows.

Theorem 2 Suppose $p>0$ . Then G is a dominanting set of ${\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)$ if and only if there are constant $\delta >0$ and $0<\eta <1$ such that

$A_{\alpha }\left(G\cap \Delta \left(a\mathrm{,}\eta \right)\right)>\delta A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)$ (2)

for all set $\Delta \left(a\mathrm{,}\eta \right)$ and all $a\in \mathbb{D}$ .

In Section 2, we mainly give several key lemmas which can prove the main result. In Section 3, we prove the main theorem by using the lemma obtained in Section 2. Section 4 gives the conclusions of this paper and explains how to extend these results to other directions.

2. Preliminaries

In this section we collect several technical lemmas that we will need for the proof of our main result. We used the convention that the letter C denotes a constant which may differ from one occurrence to the next.

Lemma 3 (Exercise 1.1.3 (b) in) Let $\mu$ be a Borel measure with $\mu \left(X\right)=1$ . We have

${\left({\displaystyle {\int }_X}{\left|f\left(x\right)\right|}^p\text{d}\mu \left(x\right)\right)}^{1/p}\ge \exp \left({\displaystyle {\int }_X}\log \left|f\left(x\right)\right|\text{d}\mu \left(x\right)\right)\mathrm{.}$

Lemma 4 (Lemma 1.24 in) For any real $\alpha$ and positive r there is constant $C>0$ and $c>0$ such that

$c{\left(1-{\left|z\right|}^2\right)}^{2+\alpha }\le A_{\alpha }\left(\Delta \left(z\mathrm{,}r\right)\right)\le C{\left(1-{\left|z\right|}^2\right)}^{2+\alpha }$

for all $z\in \mathbb{D}$ .

Lemma 5 (Lemma 2.20 in) For each $r>0$ there is a positive constant $C_r$ such that

$C_r^{-1}\le \frac{1-{\left|a\right|}^2}{1-{\left|z\right|}^2}\le C_r$

and

$C_r^{-1}\le \frac{1-{\left|a\right|}^2}{\left|1-a\bar{z}\right|}\le C_r$

for all a and z in $\mathbb{D}$ with $\rho \left(a,z\right)<r$ . Moreover, if r is bounded above, then we may choose $C_r$ to be independent of r.

Lemma 6 (Corollary 2.21 in) Suppose $-\infty <\alpha <\infty$ , $r_1>0$ , $r_2>0$ and $r_3>0$ . Then there is a constant $C>0$ such that

$C^{-1}\le \frac{A_{\alpha }\left(\Delta \left(z\mathrm{,}{r}_1\right)\right)}{A_{\alpha }\left(\Delta \left(w\mathrm{,}{r}_2\right)\right)}\le C$

for all z and w in $\mathbb{D}$ with $\rho \left(z\mathrm{,}w\right)\le r_3$ .

Lemma 7 (Lemma 2.24 in) Suppose $r>0$ , $p>0$ and $\alpha >-1$ . Then there is a constant $C>0$ such that

${\left|f\left(z\right)\right|}^p\le \frac{C}{{\left(1-{\left|z\right|}^2\right)}^{2+\alpha }}{\displaystyle {\int }_{\Delta \left(z\mathrm{,}r\right)}}{\left|f\left(w\right)\right|}^p\text{d}{A}_{\alpha }(w)$

for all $f\in H\left(\mathbb{D}\right)$ and all $z\in \mathbb{D}$ . Moreover we can obtain

${\left|f\left(z\right)\right|}^p\le \frac{C_0}{A_{\alpha }\left(\Delta \left(z\mathrm{,}r\right)\right)}{\displaystyle {\int }_{\Delta \left(z\mathrm{,}r\right)}}{\left|f\left(w\right)\right|}^p\text{d}{A}_{\alpha }(w)$

for all $z\in \mathbb{D}$ where f is holomorphic and $C_0$ is a constant independent of f and z.

If the analytic function $f\in \mathbb{D}$ and $0<\lambda <1$ we consider the local level sets of f:

${\text{E}}_{\lambda }\left(a\right)={\text{E}}_{\lambda }\left(f\mathrm{,}a\right)=\left\{z\in \Delta \left(a\mathrm{,}\eta \right)\mathrm{<mo>\; :\; </mo>}\left|f\left(z\right)\right|\mathrm{<mo>\; >\; </mo>}\lambda \left|f\left(a\right)\right|\right\}$

and the operator

$B_{\lambda }f\left(a\right)=\frac{C_0}{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }$

where $C_0$ is in Lemma 7.

By Lemma 7, we can get a inequality

$\begin{array}{c}{\left|f\left(a\right)\right|}^p\le C_0\frac{{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\frac{{\displaystyle {\int }_{\Delta \left(a\mathrm{,}\eta \right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }}{{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }}\\ =C_0\frac{{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\frac{{\displaystyle {\int }_{\Delta \left(a\mathrm{,}\eta \right)\backslash {\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }+{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }}{{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }}\\ \le C_0\frac{{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\left(1+\frac{{\lambda }^p{\left|f\left(a\right)\right|}^p{\displaystyle {\int }_{\Delta \left(a\mathrm{,}\eta \right)\backslash {\text{E}}_{\lambda }\left(a\right)}}\text{d}{A}_{\alpha }}{{\lambda }^p{\left|f\left(a\right)\right|}^p{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}\text{d}{A}_{\alpha }}\right)\\ =C_0\frac{{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }}{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}=B_{\lambda }f(a)\end{array}$

We can use the same measure as in [5] to prove the following two Lemmas.

Lemma 8 Let f is analytic in $\mathbb{D}$ , there is a constant $C_0>0$ in Lemma 7 such that

$C_1=\{\begin{array}{l}0C_0\ge 1,\\ \log \frac{1}{C_0}0<C_0<1,\end{array}$ (3)

then

$\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\ge \frac{\log \frac{1}{{\lambda }^p}+\log \frac{1}{C_0}}{\log \frac{B_{\lambda }f\left(a\right)}{{\left|f\left(a\right)\right|}^p}+\log \frac{1}{{\lambda }^p}+C_1}\mathrm{.}$

for all $a\in \mathbb{D}$ .

Proof. Applying Lemma 7 and elementary estimates we have

$\begin{array}{c}\log {\left|f\left(a\right)\right|}^p\le \log C_0+\frac{1}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}{\displaystyle {\int }_{\Delta \left(a\mathrm{,}\eta \right)}}\log {\left|f\left(z\right)\right|}^p\text{d}{A}_{\alpha }\left(z\right)\\ =\log C_0+\frac{1}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\left[{\displaystyle {\int }_{\Delta \left(a\mathrm{,}\eta \right)\backslash {\text{E}}_{\lambda }\left(a\right)}}+{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}\right]\log {\left|f\left(z\right)\right|}^p\text{d}{A}_{\alpha }\left(z\right)\\ \le \log C_0+\left[1-\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\right]\log {\lambda }^p{\left|f\left(a\right)\right|}^p\\ +\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\frac{1}{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}\log {\left|f\left(z\right)\right|}^p\text{d}{A}_{\alpha }(z)\end{array}$

$\begin{array}{l}\le \log C_0+\left[1-\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\right]\log {\lambda }^p{\left|f\left(a\right)\right|}^p\\ +\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\log \frac{C_0}{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{\displaystyle {\int }_{{\text{E}}_{\lambda }\left(a\right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }-\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\log C_0\\ \le \log C_0+\left[1-\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\right]\log {\lambda }^p{\left|f\left(a\right)\right|}^p\\ +\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\log B_{\lambda }f\left(a\right)-\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\log C_0\end{array}$

where the last inequality follows by Lemma 3. If we subtract $\log {\left|f\left(a\right)\right|}^p$ from both sides we get

$\begin{array}{c}0\le \log C_0+\left[1-\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\right]\log {\lambda }^p{\left|f\left(a\right)\right|}^p+\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\log B_{\lambda }f\left(a\right)\\ +\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\log \frac{1}{C_0}-\log {\left|f\left(a\right)\right|}^p\\ \le \log C_0+\left[1-\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\right]\log {\lambda }^p+\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\log \frac{B_{\lambda }f\left(a\right)}{{\left|f\left(a\right)\right|}^p}\\ +\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\log \frac{1}{C_0}\mathrm{.}\end{array}$

Then we have

$-\log C_0-\log {\lambda }^p\le \frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\left(\log \frac{1}{{\lambda }^p}+\log \frac{B_{\lambda }f\left(a\right)}{{\left|f\left(a\right)\right|}^p}+\log \frac{1}{C_0}\right)\mathrm{.}$

We notice that $\log {\lambda }^p<0$ , $\log \left(\frac{B_{\lambda }f\left(a\right)}{{\left|f\left(a\right)\right|}^p}\right)>0$ and $\log \frac{1}{C_0}\le C_1$ . Then we get

$\frac{A_{\alpha }\left({\text{E}}_{\lambda }\left(a\right)\right)}{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}\ge \frac{\log \frac{1}{{\lambda }^p}+\log \frac{1}{C_0}}{\log \frac{B_{\lambda }f\mathrm{<mo\; stretchy="false">\; (\; </mo>}a\mathrm{<mo\; stretchy="false">\; )\; </mo>}}{{\left|f\left(a\right)\right|}^p}+\log \frac{1}{{\lambda }^p}+C_1}\mathrm{.}$

Lemma 9 Suppose $\alpha >-1$ and $f\in {\mathcal{A}}_{\alpha }^p\left(\mathbb{D}\right)$ . Then let

$A=\left\{a\in \mathbb{D}\mathrm{<mo>\; :\; </mo>}{\left|f\left(a\right)\right|}^p<\frac{\epsilon }{A_{\alpha }\left(\Delta \left(a\mathrm{,}\eta \right)\right)}{\displaystyle {\int }_{\Delta \left(a\mathrm{,}\eta \right)}}{\left|f\right|}^p\text{d}{A}_{\alpha }\right\}$

for $\epsilon >0$ and $\eta \in \left(\mathrm{0,1}\right)$ . There is a constant C depending only on $\eta$ , such that

for all.

Proof. For we have

Integrate over and use Fubini’s Theorem on the right to obtain

where the second inequality above follows from Lemma 6 and the fact that.

Lemma 10 Let and for. Define the set

Then there is a constant C depending only on and p, such that

for all.

Proof. Let A be as in Lemma 9. We write

The first integral can be estimated by Lemma 9. For the second integral, we have

(4)

We need only show the inner integral is suitably bounded. The sets appeared in [5] , and Luecking proved in that paper that there is a constant, depending only on p, such that

(5)

Thus we can obtain

Combining this with inequality (5), we get

Plug this into (4) and use. We obtain

3. Proof of Main Theorem

We can now characterize a special family of reverse Carleson measures for weighted Bergman spaces with the weighted Lebesgue measure. The main results is as follows.

Theorem 11 Suppose. Then G is a dominanting set of if and only if there are constant and such that

(6)

for all set and all.

Proof. First, we proof the necessity of the Theorem. Take so that

By a change of variables, we get

Then we can have

Applying (1) to the function

we get

Since, so we have

It is easy to verify that

Combining this with the above inequality, we get

so inequality

which gives (6).

For sufficiency of the theorem,we will follow the arguments in [5] closely. For let

According to Lemma 10, we have

If we now choose small enough so that, we have

(7)

By the definition of F we have for all.

If, then. Lemma 8 can be write

For the in (6), we choose a positive integer, which implies that

(8)

Therefore, by choosing

Lemma 8 gives

So we have

It implies that

Note that

then following from (6) we have

whenever.

If so taht, then. Lemma 8 can be write

Therefore, by choosing

Lemma 8 gives

So we have

It implies that

Note that

then following from (6) and (8) we have

whenever.

Then we can get

for all constant in Lemma 7.

Hence we have

where. Integrating both side over F and using Fubini’s Theorem, we obtain

(9)

The integral in the brackets of the left-hand side can be estimated as follows:

And the right hand side of (9) can be estimated from below using (7). This yields

which proves the sufficiency of the theorem.

4. Conclusions

We proved the dominating set by using pseudo-hyperbolic metric disk and sub-mean inequality. The method of proof is to obtain the complete characterization of dominating set by applying the key lemma given in Section 2 in Section 3.

Next we will study some applications of Theorem 11. Let be a bounded measurable function on. So we want to prove that the Toeplitz operator is bounded. Using dominating set and Carleson measures, we can also study sampling measures for weighted Bergman space.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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