An integral-type operator from ${\mathcal{Q}}_k(p\text{,}q)$ spaces to Zygmund-type spaces

Abstract

We study the boundedness and compactness of an integral-type operator from ${\mathcal{Q}}_K(p\text{,}q)$ and ${\mathcal{Q}}_{K\text{,}0}(p\text{,}q)$ spaces to Zygmund-type spaces. In particular, we use two families of functions to characterize the boundedness and compactness.

Introduction

A positive continuous function $\varphi$ on $[0\text{,}1)$ is called normal, if there exist positive numbers s and $t\text{,}0<s<t$, and $\rho \in [0\text{,}1)$ such that (see [15])

$$\frac{\varphi (r)}{{(1-r)}^s}\text{is decreasing on}[\rho \text{,}1)\text{and}{\displaystyle \underset{r\to 1}{\lim }}\frac{\varphi (r)}{{(1-r)}^s}=0\text{;}$$

$$\frac{\varphi (r)}{{(1-r)}^t}\text{is increasing on}[\rho \text{,}1)\text{and}{\displaystyle \underset{r\to 1}{\lim }}\frac{\varphi (r)}{{(1-r)}^t}=\infty \text{.}$$

Let $\mathbb{D}$ be the open unit disk in the complex plane $\mathbb{C}\text{,}H(\mathbb{D})$ the class of all analytic functions on $\mathbb{D}$. For $s>0$, an $f\in H(\mathbb{D})$ is said to belong to the s-Bloch space, denoted by ${\mathcal{B}}^s$, if

$$\Vert f{\Vert }_{{\mathcal{B}}^s}=|f(0)|+{\displaystyle \underset{z\in \mathbb{D}}{\sup }}{(1-|z{|}^2)}^s|f^{\prime }(z)|<\infty \text{.}$$

${\mathcal{B}}^s$ is a Banach space with the above norm. When $s=1$, we get the classical Bloch space.

Let $\mu$ be a normal function on $[0\text{,}1)$. An $f\in H(\mathbb{D})$ is said to belong to the Zygmund-type space, denoted by ${\mathcal{Z}}_{\mu }$, if ${\sup }_{z\in \mathbb{D}}\mu (|z|)|f^{″}(z)|<\infty$. It is easy to see that ${\mathcal{Z}}_{\mu }$ is a Banach space with the norm

$$\Vert f{\Vert }_{{\mathcal{Z}}_{\mu }}=|f(0)|+|f^{\prime }(0)|+{\displaystyle \underset{z\in \mathbb{D}}{\sup }}\mu (|z|)|f^{″}(z)|\text{.}$$

When $\mu (r)=(1-r^2)$, we obtain the classical Zygmund space (see [2], [8]).

Assume that $p>0\text{,}q>-2\text{,}K:[0\text{,}\infty )\to [0\text{,}\infty )$ is a nondecreasing continuous function. An $f\in H(\mathbb{D})$ is said to belong to ${\mathcal{Q}}_K(p\text{,}q)$ space if (see, e.g., [13], [26])

$$\Vert f{\Vert }_{{\mathcal{Q}}_K(p\text{,}q)}^p=|f(0)|+{\displaystyle \underset{a\in \mathbb{D}}{\sup }}{\int }_{\mathbb{D}}|f^{\prime }(z){|}^p{(1-|z{|}^2)}^{q}K(g(z\text{,}a))\mathit{dA}(z)<\infty \text{,}$$

where dA is the normalized Lebesgue area measure in $\mathbb{D}\text{,}g(z\text{,}a)=\log \frac{1}{|{\phi }_a(z)|}\text{,}{\phi }_a(z)=\frac{a-z}{1-\overline{a}z}$. ${\mathcal{Q}}_K(p\text{,}q)$ is a Banach space under the norm $\Vert f{\Vert }_{{\mathcal{Q}}_K(p\text{,}q)}$ when $p⩾1$. An $f\in H(\mathbb{D})$ is said to belong to ${\mathcal{Q}}_{K\text{,}0}(p\text{,}q)$ space if

$${\displaystyle \underset{|a|\to 1}{\lim }}{\int }_{\mathbb{D}}|f^{\prime }(z){|}^p{(1-|z{|}^2)}^{q}K(g(z\text{,}a))\mathit{dA}(z)=0\text{.}$$

If $K(x)=x^s\text{,}s⩾0$, the space ${\mathcal{Q}}_K(p\text{,}q)$ equals to $F(p\text{,}q\text{,}s)$ (see [30]). When $p=2\text{,}q=0$, the space ${\mathcal{Q}}_K(p\text{,}q)$ equals to ${\mathcal{Q}}_K$, which was studied, for example, in [25]. Throughout the paper we assume that (see [26])

$${\int }_0^1{(1-r^2)}^{q}K(-\log r)\mathit{rdr}<\infty \text{,}$$

since otherwise ${\mathcal{Q}}_K(p\text{,}q)$ consists only of constant functions.

Let $\phi$ be an analytic self-map of $\mathbb{D}$. The composition operator, denoted by $C_{\phi }$, is defined by

$$C_{\phi }(f)(z)=f(\phi (z))\text{,}f\in H(\mathbb{D})\text{.}$$

Let $g\in H(\mathbb{D})$ and $\phi$ be an analytic self-map of $\mathbb{D}$. In [8], Li and Stević defined the generalized composition operator as follows

$$(C_{\phi }^{g}f)(z)={\int }_0^z{f}^{\prime }(\phi (\xi ))g(\xi )d\xi \text{,}f\in H(\mathbb{D})\text{,}z\in \mathbb{D}\text{.}$$

Some results of the generalized composition operator can be found, for example, in [9], [10], [16], [17], [18], [24], [28], [29].

Let $g\in H(\mathbb{D})\text{,}n$ be a nonnegative integer and $\phi$ be an analytic self-map of $\mathbb{D}$. Zhu generalized the operator $C_{\phi }^g$ and defined an integral-type operator as follows (see, e.g., [35]).

$$(C_{\phi \text{,}g}^{n}f)(z)={\int }_0^z{f}^{(n)}(\phi (\xi ))g(\xi )d\xi \text{,}z\in \mathbb{D}\text{,}f\in H(\mathbb{D})\text{.}$$

The operator $C_{\phi \text{,}g}^n$ is related to the generalized weighted composition operators or the weighted differentiation composition operator (see, e.g., [3], [22], [23], [27], [32], [33], [34]) in one dimension. In [14], Pan studied the operator $C_{\phi \text{,}g}^n$ from ${\mathcal{Q}}_K(p\text{,}q)$ to Bloch-type space. In [5], Li studied the operator $C_{\phi \text{,}g}^n$ from the Bloch space to the space ${\mathcal{Q}}_K(p\text{,}q)$ (see [24] for related results). See [6], [7], [11], [12], [19], [20], [21] for some operators on Zygmund spaces.

In this paper, we completely characterize the boundedness and compactness of the operator $C_{\phi \text{,}g}^n$ from ${\mathcal{Q}}_K(p\text{,}q)$ and ${\mathcal{Q}}_{K\text{,}0}(p\text{,}q)$ to Zygmund-type spaces.

Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation $A\asymp B$ means that there is a positive constant C such that $B/C⩽A⩽\mathit{CB}$.