The Bloch space and the dual space of a Luecking-type subspace of $A^1$

TL;DR

This paper investigates the relationship between the Bloch space and the dual space of a Luecking-type subspace of the Bergman space, providing negative answers to longstanding questions about their density and subset relations.

Contribution

It demonstrates that the Bloch space is neither dense in nor a subset of the dual space of the Luecking-type subspace, resolving two open questions from 1990.

Findings

The Bloch space is not dense in the dual space.

The little Bloch space is not a subset of the dual space.

Answers to Ghatage and Sun's questions are negative.

Abstract

Let $X$ be the dual space of a Luecking-type subspace of the Bergman space $A^1$. It is known that the Bloch space $\mathcal{B}$ is a subset of $X$. In 1990, Ghatage and Sun asked whether $\mathcal{B}$ is dense in $X$. They also asked whether the little version of $X$ is a subset of $\mathcal{B}$. In this note, based on results and methods of Girela, Pel\'aez, P\'erez-Gonz\'alez and R\"atty\"a in 2008, we answer the two questions in the negative.

Equations

$$\Vert f{\Vert }_{A^p}={\left({\int }_{\mathbb{D}}|f(z){|}^{p}dA(z)\right)}^{\frac{1}{p}}<\infty ,$$

$$f(z)=\sum _{n=1}^{\infty }\frac{c_n(1-|a_n|{)}^{(1-1/p)+\eta }}{(1-\overline{a_n}z{)}^{1+1/p+\eta }}$$

$$f(z)=\sum _{n=1}^{\infty }{a}_n{k}_{{\lambda }_n}(z),$$

$$\sum _{n=1}^{\infty }|a_n|\Vert k_{{\lambda }_n}{\Vert }_{A^1}<\infty .$$

$$k_w(z)=\frac{1}{(1-\overline{w}z{)}^2},w\in \mathbb{D},$$

$$\Vert f{\Vert }_Y=\inf \sum _{n=1}^{\infty }|a_n|\Vert k_{{\lambda }_n}{\Vert }_{A^1},$$

$$\Vert f{\Vert }_X=\underset{z\in \mathbb{D}}{\sup }|f(z)|\Vert k_z{\Vert }_{A^1}^{-1}<\infty .$$

$$\underset{|z|\to 1}{\lim }|f(z)|\Vert k_z{\Vert }_{A^1}^{-1}=0.$$

$$\underset{z\in \mathbb{D}}{\sup }(1-|z{|}^2)|f^{\prime }(z)|<\infty .$$

$$\underset{|z|\to 1}{\lim }(1-|z{|}^2)|f^{\prime }(z)|=0.$$

$$\Vert f{\Vert }_{H_{\log }^{\infty }}=\underset{z\in \mathbb{D}}{\sup }\frac{|f(z)|}{\log \frac{e}{1-|z|}}<\infty .$$

$$\Vert k_w{\Vert }_{A^1}=\{\begin{array}{l}1,w=0,\\ |w{|}^{-2}\log \frac{1}{(1-|w{|}^2)},w\ne 0.\end{array}$$

$$\underset{|w|\to 0}{\lim }|w{|}^{-2}\log \frac{1}{(1-|w{|}^2)}=1.$$

$$|||f||{|}_X=\underset{z\in \mathbb{D}}{\sup }\frac{|f(z)|}{\log \frac{e}{1-|z|}},f\in X.$$

$$\underset{|z|\to 1}{\lim }\frac{|f(z)|}{\log \frac{e}{1-|z|}}=0.$$

$$|f_1(z)|+|f_2(z)|\ge \log \frac{1}{1-|z|},z\in \mathbb{D}.$$

$$|||f_i-g_i||{|}_X=\underset{z\in \mathbb{D}}{\sup }\frac{|f_i(z)-g_i(z)|}{\log \frac{e}{1-|z|}}<\frac{1}{3},i=1,2.$$

$$|f_i(w)-g_i(w)|\le \frac{1}{3}\log \frac{e}{1-|w|},i=1,2,$$

$$|g_i(w)|\ge |f_i(w)|-\frac{1}{3}\log \frac{e}{1-|w|},w\in \mathbb{D},i=1,2.$$

$$|g_1(w)|+|g_2(w)|\ge \log \frac{1}{1-|w|}-\frac{2}{3}\log \frac{e}{1-|w|},w\in \mathbb{D}.$$

$$|g_1(w)|+|g_2(w)|\ge \frac{1}{12}\log \frac{e}{1-|w|},1-e^{-3}<|w|<1,$$

$$\frac{1}{2\pi }{\int }_0^{2\pi }\left(|g_1(r{e}^{i\theta })|+|g_2(r{e}^{i\theta })|\right)d\theta \ge \frac{1}{12}\log \frac{e}{1-r}.$$

$$\frac{1}{2\pi }{\int }_0^{2\pi }|g_j(r{e}^{i\theta })|d\theta \ge \frac{1}{24}\log \frac{e}{1-r},$$

$$\underset{r\to 1}{\overline{\lim }}\frac{\frac{1}{2\pi }{\int }_0^{2\pi }|g_j(r{e}^{i\theta })|d\theta }{{\left(\log \frac{1}{1-r}\right)}^{\frac{1}{2}}}=\infty .$$

$${\left(\frac{1}{2\pi }{\int }_0^{2\pi }|h(r{e}^{i\theta }){|}^{p}d\theta \right)}^{\frac{1}{p}}=O\left({\left(\log \frac{1}{1-r}\right)}^{\frac{1}{2}}\right),\text{mbox}asr\to 1,$$

$$|f^{\prime }(z)|+|g^{\prime }(z)|\ge \frac{1}{1-|z|}$$

$$\underset{k\to \infty }{\lim }\frac{|a_k|}{\log n_k}=0.$$

$$|a_k|=\frac{|f^{(n_k)}(0)|}{n_k!}\le \frac{1}{2\pi }{\int }_{|z|=1-\frac{1}{n_k}}\frac{|f(w)|}{|w{|}^{n_k+1}}|dw|\le \epsilon \log (e{n}_k){\left(1-\frac{1}{n_k}\right)}^{-n_k}\le 2\epsilon e\log (e{n}_k).$$

$$\frac{{\sum }_{k=0}^K|a_k|}{\log \frac{e}{1-|z|}}<\epsilon .$$

$$\underset{k}{\sup }\frac{|a_k|}{\log n_k}<\infty .$$

Full text

The work was supported by NNSF of China (No. 11371234 and No. 11571217).

1. Introduction

Let $\mathbb{D}$ be the open unit disk in the complex plan $\mathbb{C}$. Denote by $H(\mathbb{D})$ the space of analytic functions in $\mathbb{D}$. For $0<p<\infty$, a function $f\in H(\mathbb{D})$ belongs to the Bergman space $A^{p}$ if

[TABLE]

where $dA$ is the normalized area measure of $\mathbb{D}$. Coifman and Rochberg [6] established atomic decomposition theorem for the Bergman space $A^{p}$, $0<p<\infty$. In [15, p. 329], Luecking proved that there exists a sequence $\{a_{n}\}_{n=1}^{\infty}\subseteq\mathbb{D}$ such that every $f\in A^{p}$, $p>1$, can be written in the form

[TABLE]

for an arbitrary $\eta>0$ and $\{c_{n}\}\in\ell^{p}$. As pointed out in [15], the range of the parameter $\eta$ in the above formula is larger than the corresponding result in [6]. To determine the extent to which Luecking’s decomposition of Bergman spaces $A^{p}$, $1<p<\infty$, can be extended to $A^{1}$, Ghatage and Sun [9] introduced a Banach space of analytic functions, denoted by $Y$. More precisely, $Y$ is the completion (in the norm $\|.\|_{Y}$ defined later) of the set of functions $f\in A^{1}$ with the form

[TABLE]

where $a_{n}\in\mathbb{C}$, $\lambda_{n}\in\mathbb{D}$ and

[TABLE]

Here

[TABLE]

is the reproducing kernel of $A^{2}$. A norm of $Y$ is

[TABLE]

where the infimum is taken over the set of all such decompositions of $f$. From [9], $Y$ is a proper subset of $A^{1}$ and $Y$ is said to be a Luecking-type subspace of $A^{1}$.

Ghatage and Sun [9] gave a series of interesting results of the space $Y$. In particular, they described its dual and predual. Let $X$ be the space of functions $f\in H(\mathbb{D})$ with

[TABLE]

Denote by $X_{0}$ the little version of $X$. Namely, the space $X_{0}$ consists of functions $f\in X$ such that

[TABLE]

It is known that $X_{0}$ is a closed subspace of $X$. Ghatage and Sun [9] proved that the dual of $X_{0}$ can be identified with $Y$, and the dual of $Y$ can be identified with $X$. Recall that the Bloch space $\mathcal{B}$ is the set of functions $f\in H(\mathbb{D})$ for which

[TABLE]

The little Bloch space $\mathcal{B}_{0}$ consists of those functions $f\in\mathcal{B}$ with

[TABLE]

By [9], $\mathcal{B}\subsetneqq X$ and $\mathcal{B}_{0}\subsetneqq X_{0}$. See [3, 4, 8, 13] for the further study associated with $X$ or $Y$.

In [9, p. 771], Ghatage and Sun asked whether $\mathcal{B}$ is dense in $X$. In [9, p. 773], they asked whether $X_{0}$ is a subset of $\mathcal{B}$. In this note, we answer the two questions in the negative.

2. The Bloch space and the space $X$

The section is devoted to answer the two questions of Ghatage and Sun. By results and methods of Girela, Peláez, Pérez-González and Rättyä [10], the proof given here is elementary.

Denote by $H_{\log}^{\infty}$ the Banach space of functions $f\in H(\mathbb{D})$ satisfying

[TABLE]

See [10] for the study of $H_{\log}^{\infty}$. From [8] or [9], we know that

[TABLE]

Note that

[TABLE]

Then it is easy to see that $X=H_{\log}^{\infty}$. Hence another norm of $X$ can be defined by

[TABLE]

Also, $X_{0}$ is the set of functions $f\in X$ with

[TABLE]

From [9], $\mathcal{B}_{0}$ is densely contained in $X_{0}$. But the following result shows that the case for $\mathcal{B}$ and $X$ is different.

Theorem 2.1**.**

The Bloch space is not dense in the space $X$.

Proof.

By [10, Theorem 1.2], there exist two functions $f_{1}$, $f_{2}\in X$ such that

[TABLE]

Suppose $\mathcal{B}$ is dense in $X$. Then there are two functions $g_{1}$, $g_{2}\in\mathcal{B}$ satisfying

[TABLE]

Hence,

[TABLE]

for all $w\in\mathbb{D}$. Consequently,

[TABLE]

Combining this with (2.1), one gets

[TABLE]

Thus,

[TABLE]

which will produce a contradiction (see [10, p. 514]). In fact, for $1-e^{-3}<r<1$, we have

[TABLE]

Thus,

[TABLE]

where $j$ is equal to 1 or 2. This gives

[TABLE]

But a result of Clunie and MacGregor [5] or Makarov [16] asserts that

[TABLE]

for all $0<p<\infty$ and $h\in\mathcal{B}$. We see that condition (2.2) contradicts condition (2.3) with $p=1$. The proof is complete. ∎

Remark 1. The first construction of the same fashion as condition (2.1) was given by Ramey and Ullrich [18]. More precisely, Ramey and Ullrich proved that there exist $f$, $g\in\mathcal{B}$ such that

[TABLE]

for all $z\in\mathbb{D}$. In 2011, Kwon and Pavlović [12] generalized Ramey and Ullrich’s result by considering a wide class of weights. We refer to [1, 11, 14] for more results related to these constructions.

Remark 2. Choe and Lee [4, p. 162] also posed a question that is $\mathcal{B}$ dense in $X$ for the corresponding case in the unit ball of the complex space $\mathbb{C}^{n}$? Note that the results in the unit ball of $\mathbb{C}^{n}$ corresponding to conditions (2.1) and (2.3) can be found in [1, p. 400] and [7, p. 2808] respectively. The same arguments as the proof of Theorem 2.1 yield that the answer to Choe and Lee’s question is also negative.

Girela, Peláez, Pérez-González and Rättyä [10, Theorem 8.1] characterized certain lacunary series in $H_{\log}^{\infty}$ (see [17, Theorem 14] for a more general result). We get the corresponding result of $X_{0}$ as follows. As an application, we show that $X_{0}\nsubseteq\mathcal{B}$.

Theorem 2.2**.**

Let $f\in H(\mathbb{D})$ with the power series expansion $f(z)=\sum_{k=0}^{\infty}a_{k}z^{n_{k}}$ and suppose there exist $\alpha>1$ and $\beta>1$ such that $n_{k}^{\alpha}\leq n_{k+1}\leq n_{k}^{\alpha\beta}$ for all $k$. Then $f\in X_{0}$ if and only if

[TABLE]

Furthermore, the space $X_{0}$ is not a subset of the Bloch space.

Proof.

Let $f\in X_{0}$. Then for any $\varepsilon>0$, there exists a $\delta>0$, such that $|f(z)|<\varepsilon\log\frac{e}{1-|z|}$ for $\delta<|z|<1$. For this $\delta$, there exists a positive integer $N$, such that if $k>N$, then $1-\frac{1}{n_{k}}>\delta$. By Cauchy’s integral formula, one gets

[TABLE]

Thus condition (2.4) holds.

On the other hand, suppose condition (2.4) is true. Then for any $\varepsilon>0$, there exists a positive integer $K$, such that if $k>K$, then $|a_{k}|<\varepsilon\log n_{k}$. There also exists a positive constant $\eta$ such that if $\eta<|z|<1$, then

[TABLE]

It is clear that condition (2.4) implies

[TABLE]

Hence, by Girela, Peláez, Pérez-González and Rättyä [10, p. 536], we see that there exists a positive constant $C$ depending only on $\alpha$ and $\beta$, such that

[TABLE]

Therefore, for $\eta<|z|<1$, we obtain

[TABLE]

which gives

[TABLE]

Namely $f\in X_{0}$. Consequently, $f\in X_{0}$ if and only if condition (2.4) holds.

Let $h(z)=\sum_{k=1}^{\infty}kz^{2^{2^{k}}}$. Take $\alpha=2$ and $\beta=3/2$. Then $(2^{2^{k}})^{\alpha}=2^{2^{k+1}}<(2^{2^{k}})^{\alpha\beta}$ and

[TABLE]

Thus $h\in X_{0}$. It is well known (cf. [2]) that if $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\in\mathcal{B}$, then the sequence $\{b_{n}\}_{n=0}^{\infty}$ is bounded. Hence $h\not\in\mathcal{B}$. In other words, the space $X_{0}$ is not a subset of the Bloch space. The proof is complete. ∎

Acknowledgements

The authors thank the two anonymous referees very much for their corrections and helpful suggestions.