Essential Norm of Generalized Integral-Type Operator from the Fractional Cauchy Transform Space into Weighted Bloch and Dirichlet Spaces

Abstract

In this paper, we investigate the generalized integral-type operator acting between the fractional Cauchy transform space and two classical analytic function spaces: the weighted Bloch space and the weighted Dirichlet space. For the operator acting into the weighted Bloch space, we obtain two equivalent exact formulas for its operator norm. Furthermore, an estimate for its essential norm is provided, which leads to a necessary and sufficient condition for compactness. For the operator acting into the weighted Dirichlet space, we derive the exact operator norm and fully characterize its compactness.

1. Introduction

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$ and $\mathcal{H}\left(\mathbb{D}\right)$ be the set of analytic functions on $\mathbb{D}$. Here, we use some subspaces of $\mathcal{H}\left(\mathbb{D}\right)$, which are known in the literature. If a function f is analytic in $\mathbb{D}$, then the Cauchy formula gives

$$f\left(z\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathbb{T}}{\displaystyle \frac{f\left(\xi \right)}{\xi -z}}d\xi ,z\in \mathbb{D},$$

(1)

where $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}.$ The above representation extends to more general functions belonging to the fractional Cauchy transform space. For $\alpha >0$, a function f belongs to the space of fractional Cauchy transforms ${\mathcal{F}}_{\alpha }$ if it has a representation as

$$f\left(z\right)={\int }_{\mathbb{T}}{\displaystyle \frac{1}{{(1-\overline{\xi }z)}^{\alpha }}}d\mu \left(\xi \right),z\in \mathbb{D},$$

(2)

for some $\mu \in \mathcal{M}$, the space of all complex-valued Borel measures on $\mathbb{T}$ with the total variation norm. The representation (1) is a special case for $\alpha =1$ and $d\mu \left(\xi \right)={\displaystyle \frac{f\left(\xi \right)}{2\pi i}}d\xi$. The family of fractional Cauchy transforms ${\mathcal{F}}_{\alpha }$ has been studied extensively; two standard references are [1,2]. The space ${\mathcal{F}}_{\alpha }$ is a vector space with respect to the ordinary addition of functions and multiplication by complex numbers. The statement

$${\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}=\underset{\mu }{inf}\parallel \mu \parallel$$

defines a norm on ${\mathcal{F}}_{\alpha }$, which is a Banach space with this norm. Here, infimum is taken over measures in $\mathcal{M}$ for which (2) holds and $\parallel \mu \parallel$ is the total variation of $\mu$. It can be shown that for every $f\in {\mathcal{F}}_{\alpha }$, there is a $\mu \in \mathcal{M}$, such that (2) holds and ${\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}=\parallel \mu \parallel$.

Now, we define weighted Bloch and Dirichlet spaces. Suppose that $\omega$ is a weight function on $\mathbb{D}$, i.e., a positive continuous function on $\mathbb{D}$. The weighted Bloch space ${\mathcal{B}}_{\omega }$ is the space of all analytic functions in $\mathbb{D}$ for which

$${\parallel f\parallel }_{{\mathcal{B}}_{\omega }}=\left|f\left(0\right)\right|+\underset{z\in \mathbb{D}}{sup}\omega \left(z\right)\left|f^{\prime }\left(z\right)\right|<\infty .$$

The space ${\mathcal{B}}_{\omega }$ is a Banach space equipped with the above norm. For $\omega \left(z\right)={(1-|z|}^2{)}^{\alpha }$, $\alpha >0$, we obtain the Bloch-type space ${\mathcal{B}}^{\alpha }.$ See [3] for a more complete understanding of these spaces. For a radial weight $\omega$, $\omega \left(\right|z\left|\right)=\omega \left(z\right)$, the weighted Dirichlet space ${\mathcal{D}}_{\omega }$ is defined as follows:

$${\mathcal{D}}_{\omega }=\{f\in \mathcal{H}\left(\mathbb{D}\right):{\int }_{\mathbb{D}}{|f^{\prime }\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<\infty \}.$$

Here, $dA$ is the normalized area measure on $\mathbb{D},$ ${\int }_{\mathbb{D}}dA\left(z\right)=1$. The norm on the space ${\mathcal{D}}_{\omega }$ is

$${\parallel f\parallel }_{{\mathcal{D}}_{\omega }}=\left|f\left(0\right)\right|+{\left({\int }_{\mathbb{D}}{\left|f^{\prime }\left(z\right)\right|}^2\omega \left(z\right)dA\left(z\right)\right)}^{1/2}.$$

The classical Dirichlet space $\mathcal{D}$ is a special case of this space for $\omega \equiv 1$.

Let $\phi \in S\left(\mathbb{D}\right)$, which is the space of all analytic self-maps of $\mathbb{D}$. The composition operator induced by $\phi$ is denoted by $C_{\phi }$ and is defined by

$$C_{\phi }f=f\circ \phi f\in \mathcal{H}\left(\mathbb{D}\right).$$

The above operator can be generalized using an integral called a generalized integral type operator $C_{\phi ,g}^n$ or a generalized integration operator. For $g\in \mathcal{H}\left(\mathbb{D}\right)$, $n\in {\mathbb{N}}_0=\{0,1,\cdots \}$, and $\phi \in S\left(\mathbb{D}\right)$, the generalized integral-type operator is defined as follows:

$$\left(C_{\phi ,g}^{n}f\right)\left(z\right)={\int }_0^z{f}^{\left(n\right)}\left(\phi \left(\zeta \right)\right)g\left(\zeta \right)d\zeta ,f\in \mathcal{H}\left(\mathbb{D}\right),z\in \mathbb{D}.$$

If $n=1$, then $C_{\phi ,g}^n=C_{\phi }^g$. This kind of operator also includes integral and Volterra operators. The generalized integration operator between the Bloch type space and weighted Dirichlet type spaces was investigated in [4]. For studies on composition operators on the space of fractional Cauchy transforms, see [5,6]. For differences in composition operators and differentiation composition operators on fractional Cauchy transforms spaces, one can refer to [7,8]. The boundedness, essential norm, and compactness of generalized Stević–Sharma-type operators on spaces of fractional Cauchy transforms have been characterized in [9]. For additional studies on this type of operator or other generalization of composition operators on spaces of analytic function spaces, especially the spaces we consider here, see [10,11]. To the best of our knowledge, the operator $C_{\phi ,g}^n$ on ${\mathcal{F}}_{\alpha }$ has not been systematically studied before. The main novelties of this paper are the following: (i) exact norm formulas for both target spaces, (ii) an essential norm estimate (equivalence) for the Bloch case, and (iii) a compactness criterion for the Dirichlet case.

In this paper, we compute or estimate two features of $C_{\phi ,g}^n$ as a linear operator, norm, and essential norm. First, we find two formulas for the norm of $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$, and as an application, the bounded operators are characterized. Then, an estimate for the essential norm is proved. Finally, we find the norm of the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$ and characterize its compactness.

For a bounded linear operator $T:X\to Y$ between two Banach spaces, the essential norm ${\parallel T\parallel }_e$ is the distance from T to the space of all compact operators from X into Y. This means that

$${\parallel T\parallel }_e=inf\{\parallel T-K\parallel :K:X\to Y\mathrm{is}\mathrm{compact}\}.$$

Therefore, the operator T is compact if and only if ${\parallel T\parallel }_e=0$.

In this work, if there exists a constant C such that $A\le CB$, we say that $A⪯B$. The symbol $A\approx B$ means that $A⪯B⪯A$.

2. Norm and Essential Norm of ${\mathit{C}}_{\mathsf{\phi }\mathbf{,}\mathit{g}}^{\mathit{n}}\mathbf{:}{\mathcal{F}}_{\mathsf{\alpha }}\mathbf{\to }{\mathcal{B}}_{\mathsf{\omega }}$

In this section, we obtain two exact formulas for the norm of the generalized integral-type operator from the Cauchy transform space into a weighted Bloch space. We also find an estimate for the essential norm of the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$. As a result, we present equivalence conditions for the compactness of such operators. First, we state the following estimate of $|f^{\left(n\right)}\left(z\right)|$ for functions in ${\mathcal{F}}_{\alpha }$:

Lemma 1.

If $f\in {\mathcal{F}}_{\alpha }$ and $n\in {\mathbb{N}}_0$, then

$$|f^{\left(n\right)}\left(z\right)|\le {\displaystyle \frac{{C\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}}{{(1-|z|}^2{)}^{\alpha +n}}},z\in \mathbb{D}$$

(3)

where C is a positive constant independent of f.

Proof. 

Since $f\in {\mathcal{F}}_{\alpha }$, there exists a $\mu \in \mathcal{M}$ such that

$$f\left(z\right)={\int }_{\mathbb{T}}{\displaystyle \frac{1}{{(1-\overline{\xi }z)}^{\alpha }}}d\mu \left(\xi \right).$$

Taking the derivative of the above equation, we have

$$f^{\left(n\right)}\left(z\right)={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l){\int }_{\mathbb{T}}{\displaystyle \frac{{\overline{\xi }}^n}{{(1-\overline{\xi }z)}^{\alpha +n}}}d\mu \left(\xi \right).$$

Thus,

$$|f^{\left(n\right)}\left(z\right)|\le C{\int }_{\mathbb{T}}{\displaystyle \frac{d\left|\mu \right|\left(\xi \right)}{|1-\overline{\xi }{z|}^{\alpha +n}}}\le C{\int }_{\mathbb{T}}{\displaystyle \frac{d\left|\mu \right|\left(\xi \right)}{{(1-|z|}^2{)}^{\alpha +n}}}={\displaystyle \frac{C\parallel \mu \parallel }{{(1-|z|}^2{)}^{\alpha +n}}}.$$

Taking the infimum over all measures in $\mathcal{M}$, inequality (3) is obtained.   □

In the following theorem, we obtain the first formula for the norm of the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$.

Theorem 1.

Let $n\in {\mathbb{N}}_0$, $\alpha >0$, ω be a weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then,

$$\begin{array}{c}\hfill \parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }}={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}.\end{array}$$

(4)

Proof. 

For any $\xi \in \mathbb{T}$, let $f_{\xi }\left(z\right)={\displaystyle \frac{1}{{(1-\overline{\xi }z)}^{\alpha }}}$. Then, $f_{\xi }\in {\mathcal{F}}_{\alpha }$, $\parallel f_{\xi }{\parallel }_{{\mathcal{F}}_{\alpha }}=1$, and

$$f_{\xi }^{\left(k\right)}\left(z\right)={\displaystyle \frac{{\overline{\xi }}^k{\prod }_{l=0}^{k-1}(\alpha +l)}{{(1-\overline{\xi }z)}^{\alpha +k}}},$$

where $k\in \mathbb{N}$ [12]. The definition of the norm of the operator implies that

$$\begin{array}{cc}\hfill \parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }}& \ge \parallel C_{\phi ,g}^n\left(f_{\xi }\right){\parallel }_{{\mathcal{B}}_{\omega }}\hfill \\ \hfill & =|C_{\phi ,g}^n\left(f_{\xi }\right)\left(0\right)|+\underset{z\in \mathbb{D}}{sup}\omega \left(z\right)|{\left(C_{\phi ,g}^n{f}_{\xi }\right)}^{\prime }\left(z\right)|\hfill \end{array}$$

(5)

Since $C_{\phi ,g}^n\left(f_{\xi }\right)\left(0\right)=0$ and ${\left(C_{\phi ,g}^n{f}_{\xi }\right)}^{\prime }\left(z\right)=g\left(z\right)f_{\xi }^{\left(n\right)}\left(\phi \left(z\right)\right)$, we obtain

$$\parallel C_{\phi ,g}^n\left(f_{\xi }\right){\parallel }_{{\mathcal{B}}_{\omega }}={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{z\in \mathbb{D}}{sup}·{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}.$$

Taking the supremum over $\xi \in \mathbb{T}$ gives

$$\parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }}\ge {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}.$$

Conversely, for any $f\in {\mathcal{F}}_{\alpha }$, there exists $\nu \in \mathcal{M}$ such that $\parallel \nu \parallel ={\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}$ and

$$f\left(z\right)={\int }_{\mathbb{T}}{\displaystyle \frac{d\nu \left(\xi \right)}{{(1-\overline{\xi }z)}^{\alpha }}};$$

see [12]. Moreover,

$$f^{\left(n\right)}\left(z\right)={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l){\int }_{\mathbb{T}}{\displaystyle \frac{{\overline{\xi }}^n}{{(1-\overline{\xi }z)}^{n+\alpha }}}d\nu \left(\xi \right).$$

Hence, for any $f\in {\mathcal{F}}_{\alpha }$, we have

$$\begin{array}{cc}\hfill \parallel C_{\phi ,g}^n{f\parallel }_{{\mathcal{B}}_{\omega }}& = |\left(C_{\phi ,g}^{n}f\right)\left(0\right)|+\underset{z\in \mathbb{D}}{sup}\omega \left(z\right)|{\left(C_{\phi ,g}^{n}f\right)}^{\prime }\left(z\right)|\hfill \\ \hfill & ={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{z\in \mathbb{D}}{sup}\omega \left(z\right)|g\left(z\right)|\left|{\int }_{\mathbb{T}}{\displaystyle \frac{{\overline{\xi }}^n}{{(1-\overline{\xi }\phi \left(z\right))}^{n+\alpha }}}d\nu \left(\xi \right)\right|\hfill \\ \hfill & \le {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{z\in \mathbb{D}}{sup}\omega \left(z\right)|g\left(z\right)|{\int }_{\mathbb{T}}{\displaystyle \frac{1}{|1-\overline{\xi }\phi \left(z\right){)|}^{n+\alpha }}}d|\nu |\left(\xi \right)\hfill \\ \hfill & \le {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}{\int }_{\mathbb{T}}d|\nu |\left(\xi \right)\hfill \\ \hfill & \le {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}\parallel \nu \parallel \hfill \\ \hfill & ={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}{\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\hfill \\ \hfill & =\underset{\xi \in \mathbb{T}}{sup}\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{{\prod }_{l=0}^{n-1}(\alpha +l)\omega \left(z\right)|g\left(z\right)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}{\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}.\hfill \end{array}$$

Considering the above discussion with regard to inequality (5) yields (4).   □

In the next theorem, we find another formula for the norm of the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$.

Theorem 2.

Let $n\in {\mathbb{N}}_0$, $\alpha >0$, ω be a weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then,

$$\begin{array}{c}\hfill \parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }}={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-|\phi \left(z\right)|)}^{n+\alpha }}}.\end{array}$$

Proof. 

For any $w\in \mathbb{D}$, there exists $0\le {\theta }_w<2\pi$ such that $\phi \left(w\right)=e^{i{\theta }_w}|\phi \left(w\right)|$. Thus, for any $w\in \mathbb{D}$, using the previous theorem, we have

$$\begin{array}{cc}\hfill \parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }}& ={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}\hfill \\ \hfill & \ge {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}{\displaystyle \frac{\omega \left(w\right)|g(w)|}{|1-\overline{\xi }{\phi \left(w\right)|}^{n+\alpha }}}\hfill \\ \hfill & \ge {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l){\displaystyle \frac{\omega \left(w\right)|g(w)|}{|1-\overline{e^{i{\theta }_w}}{\phi \left(w\right)|}^{n+\alpha }}}\hfill \\ \hfill & ={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l){\displaystyle \frac{\omega \left(w\right)|g(w)|}{|1-\overline{e^{i{\theta }_w}}{e}^{i{\theta }_w}{|\phi \left(w\right)||}^{n+\alpha }}}\hfill \\ \hfill & ={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l){\displaystyle \frac{\omega \left(w\right)|g(w)|}{{(1-|\phi \left(w\right)|)}^{n+\alpha }}}.\hfill \end{array}$$

By taking the supremum on w, we obtain

$$\begin{array}{c}\hfill \parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }}\ge {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(w\right)|g(w)|}{{(1-|\phi \left(w\right)|)}^{n+\alpha }}}\end{array}$$

Also, by Theorem 1, we have

$$\begin{array}{c}\hfill \parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }}={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}\le {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{z\in \mathbb{D}}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-|\phi \left(z\right)|)}^{n+\alpha }}}.\end{array}$$

The proof is complete.   □

From Theorem 1 and 2, we obtain the following corollary.

Corollary 1.

Let $n\in {\mathbb{N}}_0$, $\alpha >0$, ω be a weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then, the following statements are equivalent:

1. 

The operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$ is bounded.

2. 

${sup}_{z\in \mathbb{D}}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-|\phi \left(z\right)|)}^{n+\alpha }}}<\infty .$

3. 

${sup}_{\xi \in \mathbb{T}}{sup}_{z\in \mathbb{D}}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{|1-\overline{\xi }{\phi \left(z\right)|}^{n+\alpha }}}<\infty .$

In the following theorem, we obtain an estimate for the essential norm of the generalized integral-type operator from the Cauchy transform space into a weighted Bloch space.

Theorem 3.

Let $n\in {\mathbb{N}}_0$, $\alpha >0$, ω be a weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. If $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$ is bounded, then

$$\parallel C_{\phi ,g}^n{\parallel }_{ϵ}\approx \underset{|\phi (z)|\to 1}{lim\; sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-|\phi \left(z\right)|}^2{)}^{\alpha +n}}}.$$

Proof. 

It is clear that when ${\parallel \phi \parallel }_{\infty }={sup}_{z\in \mathbb{D}}|\phi \left(z\right)|<1$, then the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$ is compact and both sides of the desired equality are zero. Hence, we assume ${\parallel \phi \parallel }_{\infty }=1$. Choose $\left\{z_k\right\}\subseteq \mathbb{D}$ such that ${lim}_{k\to \infty }\left|\phi \left(z_k\right)\right|=1$. Consider the functions $\left\{f_k\right\}$,

$$f_{k,\phi \left(z_k\right)}\left(z\right)={\displaystyle \frac{(1-|\phi \left(z_k\right){{|}^2)}^{\alpha }}{{(1-\overline{\phi \left(z_k\right)}z)}^{\alpha +k}}}.$$

Then, $\left\{f_{k,\phi \left(z_k\right)}\right\}$ is a bounded sequence in ${\mathcal{F}}_{\alpha }$ that converges to zero uniformly on compact subsets of $\mathbb{D}$ (see [12]). Thus, using Lemma 2.10 of [13] for any compact operator $T:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$, we get

$$\begin{array}{cc}\hfill \parallel C_{\phi ,g}^n-T\parallel & ⪰\underset{k\to \infty }{lim\; sup}{\parallel (C_{\phi ,g}^n-T)f_{k,\phi \left(z_k\right)}\parallel }_{{\mathcal{B}}_{\omega }}\hfill \\ & \ge \underset{k\to \infty }{lim\; sup}(\parallel C_{\phi ,g}^n{f}_{k,\phi \left(z_k\right)}{\parallel }_{{\mathcal{B}}_{\omega }}-{\parallel T{f}_{k,\phi \left(z_k\right)}\parallel }_{{\mathcal{B}}_{\omega }})\hfill \\ \hfill & =\underset{k\to \infty }{lim\; sup}{\parallel C_{\phi ,g}^n{f}_{k,\phi \left(z_k\right)}\parallel }_{{\mathcal{B}}_{\omega }}\hfill \\ \hfill & =\underset{k\to \infty }{lim\; sup}\underset{z\in \mathbb{D}}{sup}\omega \left(z\right)|{\left(C_{\phi ,g}^n\left(f_{k,\phi \left(z_k\right)}\right)\right)}^{\prime }\left(z\right)|\hfill \\ \hfill & \ge \underset{k\to \infty }{lim\; sup}\omega \left(z_k\right)|{\left(C_{\phi ,g}^n\left(f_{k,\phi \left(z_k\right)}\right)\right)}^{\prime }\left(z_k\right)|\hfill \\ \hfill & =\underset{k\to \infty }{lim\; sup}\omega \left(z_k\right)|f_{k,\phi \left(z_k\right)}^{\left(n\right)}\left(\phi \left(z_k\right)\right)||g\left(z_k\right)|\hfill \\ \hfill & ={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{k\to \infty }{lim\; sup}{\displaystyle \frac{\omega \left(z_k\right)|\phi \left(z_k\right){|}^n|g\left(z_k\right)|}{(1-|\phi \left(z_k\right){{|}^2)}^{\alpha +n}}}.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill \parallel C_{\phi ,g}^n{\parallel }_e=\underset{T\mathrm{is}\mathrm{compact}}{inf}\parallel C_{\phi ,g}^n-T\parallel & ⪰\underset{k\to \infty }{lim\; sup}{\displaystyle \frac{\omega \left(z_k\right)|\phi \left(z_k\right){|}^n|g\left(z_k\right)|}{(1-|\phi \left(z_k\right){{|}^2)}^{\alpha +n}}}\hfill \\ \hfill & =\underset{|\phi (z)|\to 1}{lim\; sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-|\phi \left(z\right)|}^2{)}^{\alpha +n}}}.\hfill \end{array}$$

For the upper estimate, consider the operators $K_r$ on ${\mathcal{F}}_{\alpha }$, $K_{r}f\left(z\right)=f_r\left(z\right)=f\left(rz\right)$, where $0<r<1$. Then, $K_r$ is a compact operator on ${\mathcal{F}}_{\alpha }$. Let $\left\{r_j\right\}\subset (0,1)$ be an increasing sequence such that $r_j\to 1$ as $j\to \infty$. As $j\to \infty$, $f_{r_j}\to f$ uniformly on compact subsets of $\mathbb{D}$. Therefore, for any positive integer j, the operator $C_{\phi ,g}^n{K}_{r_j}:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$ is compact. Let $f\in {\mathcal{F}}_{\alpha }$ with ${\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\le 1$. Then,

$$\begin{array}{cc}\hfill \parallel (C_{\phi ,g}^n-C_{\phi ,g}^n{K}_{r_j}){f\parallel }_{{\mathcal{B}}_{\omega }}=& \underset{z\in \mathbb{D}}{sup}\omega \left(z\right)|{\left((C_{\phi ,g}^n-C_{\phi ,g}^n{K}_{r_j})f\right)}^{\prime }\left(z\right)|\hfill \\ \hfill =& \underset{z\in \mathbb{D}}{sup}\omega \left(z\right)|f^{\left(n\right)}\left(\phi \left(z\right)\right)-f^{\left(n\right)}\left(r_j\phi \left(z\right)\right)||g\left(z\right)|\hfill \\ \hfill =& \underset{|\phi \left(z\right)|\le r_N}{sup}\omega \left(z\right)|f^{\left(n\right)}\left(\phi \left(z\right)\right)-f^{\left(n\right)}\left(r_j\phi \left(z\right)\right)||g\left(z\right)|\hfill \\ \hfill & +\underset{r_N<|\phi \left(z\right)|<1}{sup}\omega \left(z\right)|f^{\left(n\right)}\left(\phi \left(z\right)\right)-f^{\left(n\right)}\left(r_j\phi \left(z\right)\right)||g\left(z\right)|,\hfill \end{array}$$

(6)

where $N\in \mathbb{N}$ is large enough such that $r_j\ge {\displaystyle \frac{1}{2}}$ for all $j\ge N$. As $f_{r_j}^{\left(n\right)}\to f^{\left(n\right)}$ uniformly on compact subsets of $\mathbb{D}$,

$$\underset{j\to \infty }{lim\; sup}\underset{|\phi \left(z\right)|\le r_N}{sup}\omega \left(z\right)|f^{\left(n\right)}\left(\phi \left(z\right)\right)-f^{\left(n\right)}\left(r_j\phi \left(z\right)\right)||g\left(z\right)|=0.$$

(7)

Conversely, using Lemma 1 and Theorem 2, we get

$$\begin{array}{cc}\hfill \underset{r_N<|\phi \left(z\right)|<1}{sup}& \omega \left(z\right)|f^{\left(n\right)}\left(\phi \left(z\right)\right)-f^{\left(n\right)}\left(r_j\phi \left(z\right)\right)||g\left(z\right)|\hfill \\ \hfill \le & \underset{r_N<|\phi \left(z\right)|<1}{sup}\omega \left(z\right)(|f^{\left(n\right)}\left(\phi \left(z\right)\right)| + |f^{\left(n\right)}\left(r_j\phi \left(z\right)\right)|)|g\left(z\right)|\hfill \\ \hfill \le & \underset{r_N<|\phi \left(z\right)|<1}{sup}\omega \left(z\right)|f^{\left(n\right)}\left(\phi \left(z\right)\right)||g\left(z\right)|\hfill \\ \hfill & +\underset{r_N<|\phi \left(z\right)|<1}{sup}\omega \left(z\right)|f^{\left(n\right)}\left(r_j\phi \left(z\right)\right)||g\left(z\right)|\hfill \\ \hfill ⪯& \underset{r_N<|\phi \left(z\right)|<1}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-{|\phi (z)|}^2)}^{\alpha +n}}}{\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\hfill \\ \hfill & +\underset{r_N<|\phi \left(z\right)|<1}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-{|r_j\phi \left(z\right)|}^2)}^{\alpha +n}}}{\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\hfill \\ \hfill \le & \underset{r_N<|\phi \left(z\right)|<1}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-|\phi {\left(z\right)|}^2)}^{\alpha +n}}}+\underset{r_N<|\phi \left(z\right)|<1}{sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-|r_j\phi \left(z\right){|}^2)}^{\alpha +n}}}.\hfill \end{array}$$

Whenever $j\to \infty$, we obtain $|\phi (z)|\to 1$; hence, the above expression tends to

$$\begin{array}{c}\hfill \underset{|\phi (z)|\to 1}{lim\; sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-{|\phi (z)|}^2)}^{\alpha +n}}}.\end{array}$$

(8)

Applying relations (6)–(8), we get

$$\begin{array}{c}\hfill \parallel C_{\phi ,g}^n-C_{\phi ,g}^n{K}_{r_j}\parallel =\underset{{\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\le 1}{sup}{\parallel (C_{\phi ,g}^n-C_{\phi ,g}^n{K}_{r_j})f\parallel }_{{\mathcal{B}}_{\omega }}⪯\underset{|\phi (z)|\to 1}{lim\; sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-{|\phi (z)|}^2)}^{\alpha +n}}}.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \parallel C_{\phi ,g}^n{\parallel }_e\le \underset{j\to \infty }{lim\; sup}\parallel C_{\phi ,g}^n-C_{\phi ,g}^n{K}_{r_j}\parallel ⪯\underset{|\phi (z)|\to 1}{lim\; sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-{|\phi (z)|}^2)}^{\alpha +n}}}.\end{array}$$

The proof is complete.   □

Corollary 2.

Let $n\in {\mathbb{N}}_0$, $\alpha >0$, ω be a weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$ such that the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$ is bounded. Then, the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$ is compact if and only if

$$\underset{|\phi (z)|\to 1}{lim\; sup}{\displaystyle \frac{\omega \left(z\right)|g(z)|}{{(1-|\phi \left(z\right)|}^2{)}^{\alpha +n}}}=0.$$

3. Boundedness and Compactness of ${\mathit{C}}_{\mathsf{\phi }\mathbf{,}\mathit{g}}^{\mathit{n}}\mathbf{:}{\mathcal{F}}_{\mathsf{\alpha }}\mathbf{\to }{\mathcal{D}}_{\mathsf{\omega }}$

In this section, we investigate the norm, boundedness, and compactness of the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$. First, we obtain the exact norm of the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$.

Theorem 4.

Let $n\in \mathbb{N}$, $\alpha >0$, ω be a radial weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then,

$$\parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }}={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}{\left({\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)\right)}^{{\displaystyle \frac{1}{2}}}.$$

Proof. 

Consider the function

$$f_{\xi }\left(z\right)={\displaystyle \frac{1}{{(1-\overline{\xi }z)}^{\alpha }}},\xi \in \mathbb{T}.$$

It is clear that $\parallel f_{\xi }{\parallel }_{{\mathcal{F}}_{\alpha }}=1$. Therefore,

$$\begin{array}{cc}\hfill \parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }}\ge {\parallel C_{\phi ,g}^n{f}_{\xi }\parallel }_{{\mathcal{D}}_{\omega }}=& \underset{0}{\underbrace{|\left(C_{\phi ,g}^n{f}_{\xi }\right)\left(0\right)|}}+{\left({\int }_{\mathbb{D}}|f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill =& {\left({({\displaystyle \prod _{l=0}^{n-1}}(\alpha +l))}^2{\int }_{\mathbb{D}}{\displaystyle \frac{{|\xi |}^{2n}}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(\alpha +n)}}}{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill =& {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l){\left({\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(\alpha +n)}}}\omega \left(z\right)dA\left(z\right)\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

The above holds for every $\xi \in \mathbb{T}$. Thus,

$${\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}{\left({\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(\alpha +n)}}}\omega \left(z\right)dA\left(z\right)\right)}^{{\displaystyle \frac{1}{2}}}\le {\parallel C_{\phi ,g}^n\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }}.$$

Now, we prove the other part of the inequality. Let $f\in {\mathcal{F}}_{\alpha }$. Then, there exists a $\mu \in \mathcal{M}$ such that $\parallel \mu \parallel ={\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}$ and

$$f\left(z\right)={\int }_{\mathbb{T}}{\displaystyle \frac{d\mu \left(\xi \right)}{{(1-\overline{\xi }z)}^{\alpha }}}.$$

Moreover,

$$f^{\left(n\right)}\left(z\right)={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l){\int }_{\mathbb{T}}{\displaystyle \frac{{\overline{\xi }}^n}{{(1-\overline{\xi }z)}^{n+\alpha }}}d\mu \left(\xi \right).$$

Thus, Jensen’s inequality implies that

$$\begin{array}{cc}\hfill |f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2& {|g\left(z\right)|}^2\omega \left(z\right)\hfill \\ \hfill & ={\left({\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\right)}^2{\left|{\int }_{\mathbb{T}}{\displaystyle \frac{{\overline{\xi }}^n}{{(1-\overline{\xi }\phi \left(z\right))}^{n+\alpha }}}d\mu \left(\xi \right)\right|}^2{|g\left(z\right)|}^2\omega \left(z\right)\hfill \\ \hfill & \le {\left({\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\right)}^2{\parallel \mu \parallel }^2{\int }_{\mathbb{T}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right){\displaystyle \frac{d|\mu |\left(\xi \right)}{\parallel \mu \parallel }}.\hfill \end{array}$$

Set $\beta ={\prod }_{l=0}^{n-1}(\alpha +l)$. Then, based on Fubini’s Theorem,

$$\begin{array}{cc}\hfill {\int }_{\mathbb{D}}|f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\le & {\beta }^2\parallel \mu \parallel {\int }_{\mathbb{D}}{\int }_{\mathbb{T}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)d|\mu |\left(\xi \right)dA\left(z\right)\hfill \\ \hfill =& {\beta }^2\parallel \mu \parallel {\int }_{\mathbb{T}}{\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)d|\mu |\left(\xi \right)\hfill \\ \hfill \le & {\beta }^2\parallel \mu \parallel \underset{\xi \in \mathbb{T}}{sup}{\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right){\int }_{\mathbb{T}}d|\mu |\left(\xi \right)\hfill \\ \hfill \le & {\beta }^2{\parallel \mu \parallel }^2\underset{\xi \in \mathbb{T}}{sup}{\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill =& {\beta }^2{\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}^2\underset{\xi \in \mathbb{T}}{sup}{\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \parallel C_{\phi ,g}^n{f\parallel }_{{\mathcal{D}}_{\omega }}=& {\left({\int }_{\mathbb{D}}|f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill \le & {\beta \parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\underset{\xi \in \mathbb{T}}{sup}{\left({\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

Therefore,

$$\parallel C_{\phi ,g}^n{\parallel }_{{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }}\le {\displaystyle \prod _{l=0}^{n-1}}(\alpha +l)\underset{\xi \in \mathbb{T}}{sup}{\left({\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)\right)}^{{\displaystyle \frac{1}{2}}}.$$

The proof is complete.   □

Through the application of Theorem 4, we obtain the following corollary.

Corollary 3.

Let $n\in \mathbb{N}$, $\alpha >0$, ω be a radial weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then, the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$ is bounded if and only if

$$\underset{\xi \in \mathbb{T}}{sup}{\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)<\infty .$$

Corollary 4.

Let $n\in \mathbb{N}$, $\alpha >0$, ω be a radial weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. If

$${\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{{(1-|\phi \left(z\right)|)}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)<\infty$$

then $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$ is bounded.

Proof. 

For any $\xi \in \mathbb{T}$,

$${\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)\le {\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{{(1-|\phi \left(z\right)|)}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right).$$

Therefore,

$$\underset{\xi \in \mathbb{T}}{sup}{\int }_{\mathbb{D}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)<\infty .$$

Hence, based on Corollary 3, the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$ is bounded.   □

Now, we find the compactness criteria of the operator.

Theorem 5.

Let $n\in \mathbb{N}$, $\alpha >0$, ω be a radial weight, $g\in \mathcal{H}\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$ such that the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$ is bounded. Then, the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$ is compact if and only if

$$\underset{r\to 1}{lim}\underset{\xi \in \mathbb{T}}{sup}{\int }_{|\phi (z)|>r}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)=0.$$

(9)

Proof. 

First, assume that (9) holds. To prove compactness, we show that for every bounded sequence ${\left\{f_j\right\}}_{j\in \mathbb{N}}$ in ${\mathcal{F}}_{\alpha }$, which converges to zero uniformly on compact subsets of $\mathbb{D}$, ${lim}_{j\to \infty }{\parallel C_{\phi ,g}^n{f}_j\parallel }_{{\mathcal{D}}_{\omega }}=0$; see Lemma 2.10 [13]. Since for each $j\in \mathbb{N}$, $f_j\in {\mathcal{F}}_{\alpha }$, there exists ${\mu }_j\in \mathcal{M}$ such that $\parallel {\mu }_j\parallel =\parallel f_j{\parallel }_{{\mathcal{F}}_{\alpha }}$ and

$$f_j\left(z\right)={\int }_{\mathbb{T}}{\displaystyle \frac{d{\mu }_j\left(\xi \right)}{{(1-\overline{\xi }z)}^{\alpha }}}.$$

In this case, we have

$$f_j^{\left(n\right)}\left(z\right)={\displaystyle \prod _{l=0}^{n-1}}(\alpha +l){\int }_{\mathbb{T}}{\displaystyle \frac{{\overline{\xi }}^n}{{(1-\overline{\xi }z)}^{n+\alpha }}}d{\mu }_j\left(\xi \right).$$

Set $\beta ={\prod }_{l=0}^{n-1}(\alpha +l).$ Applying Jensen’s inequality, we have

$$\begin{array}{cc}\hfill |f_j^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)=& {\beta }^2{\left|{\int }_{\mathbb{T}}{\displaystyle \frac{{\overline{\xi }}^n}{{(1-\overline{\xi }\phi \left(z\right))}^{n+\alpha }}}d{\mu }_j\left(\xi \right)\right|}^2{|g\left(z\right)|}^2\omega \left(z\right)\hfill \\ \hfill \le & {\beta }^2{\parallel {\mu }_j\parallel }^2{\int }_{\mathbb{T}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right){\displaystyle \frac{d|{\mu }_j|\left(\xi \right)}{\parallel {\mu }_j\parallel }}.\hfill \end{array}$$

(10)

It follows from (9) that for every $ϵ>0$, we can find $0<r_1<1$ such that for any $r\in (r_1,1)$,

$$\underset{\xi \in \mathbb{T}}{sup}{\int }_{|\phi (z)|>r}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)<ϵ.$$

(11)

Therefore,

$$\begin{array}{cc}\hfill \parallel C_{\phi ,g}^n{f}_j{\parallel }_{{\mathcal{D}}_{\omega }}^2=& {\int }_{\mathbb{D}}|f_j^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill \le & {\int }_{|\phi (z)|\le r}|f_j^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill & +{\int }_{|\phi (z)|>r}|f_j^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill & :=Q_1+Q_2.\hfill \end{array}$$

(12)

We know that $f_j^{\left(n\right)}\to 0$ uniformly on compact subsets of $\mathbb{D}.$ Therefore,

$$\underset{|\phi (z)|\le r}{sup}{|f_j^{\left(n\right)}\left(\phi \left(z\right)\right)|}^2<ϵ,$$

for sufficiently large j, say $j\ge j_0$. In this case

$$\begin{array}{c}\hfill Q_1={\int }_{|\phi (z)|\le r}|f_j^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<ϵ{\int }_{\mathbb{D}}{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)⪯ϵ.\end{array}$$

(13)

Note that we use the fact that ${\int }_{\mathbb{D}}{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<\infty$, which can be obtained from the boundedness of the operator $C_{\phi ,g}^n$, applied to the function $f\left(z\right)=z^n$.

• Moreover, Equations (10) and (11) imply that

  $$\begin{array}{cc}\hfill Q_2={\int }_{|\phi (z)|>r}& |f_j^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill \le & {\beta }^2\parallel {\mu }_j\parallel {\int }_{|\phi (z)|>r}{\int }_{\mathbb{T}}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)d|{\mu }_j|\left(\xi \right)dA\left(z\right)\hfill \\ \hfill =& {\beta }^2\parallel {\mu }_j\parallel {\int }_{\mathbb{T}}{\int }_{|\phi (z)|>r}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)d|{\mu }_j|\left(\xi \right)\hfill \\ \hfill \le & {\beta }^2\parallel {\mu }_j\parallel \underset{\xi \in \mathbb{T}}{sup}{\int }_{|\phi (z)|>r}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right){\int }_{\mathbb{T}}d|{\mu }_j|\left(\xi \right)\hfill \\ \hfill \le & {\beta }^2{\parallel {\mu }_j\parallel }^2\underset{\xi \in \mathbb{T}}{sup}{\int }_{|\phi (z)|>r}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill ⪯& ϵ.\hfill \end{array}$$

  (14)

  From Equations (12)–(14), we get $\parallel C_{\phi ,g}^n{f}_j{\parallel }_{{\mathcal{D}}_{\omega }}⪯\sqrt{2ϵ}.$ Hence, the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$ is compact.

To prove the converse, note that ${\int }_{\mathbb{D}}{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<\infty$ by applying the bounded operator $C_{\phi ,g}^n$ to the function $f\left(z\right)=z^n$. Thus, for each $ϵ>0$, there is an $r\in (0,1)$ such that

$${\int }_{|\phi (z)|>r}{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<ϵ.$$

(15)

Consider the functions $f_t\left(z\right)=f\left(tz\right)$, $0<t<1$, and $f\in {\mathcal{F}}_{\alpha }$. Then, ${sup}_{0<t<1}\parallel f_t{\parallel }_{{\mathcal{F}}_{\alpha }}\le {\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}$ and $f_t\to f$ uniformly on compact subsets of $\mathbb{D}$ as $t\to 1$. From the compactness of the operator $C_{\phi ,g}^n$ and Lemma 2.10 of [13], we get

$$\underset{t\to 1}{lim}{\parallel C_{\phi ,g}^n{f}_t-C_{\phi ,g}^{n}f\parallel }_{{\mathcal{D}}_{\omega }}=0.$$

Thus, for each $ϵ>0$, there is a $t\in (0,1)$ such that

$$\parallel C_{\phi ,g}^n{f}_t-C_{\phi ,g}^n{f\parallel }_{{\mathcal{D}}_{\omega }}^2={\int }_{\mathbb{D}}|f_t^{\left(n\right)}\left(\phi \left(z\right)\right)-f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<ϵ.$$

(16)

From (15) and (16), we have

$$\begin{array}{cc}\hfill {\int }_{|\phi (z)|>r}& |f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill \le & {\int }_{|\phi (z)|>r}|f_t^{\left(n\right)}\left(\phi \left(z\right)\right)-f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill & +{\int }_{|\phi (z)|>r}|f_t^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)\hfill \\ \hfill \le & ϵ+{\parallel f_t^{\left(n\right)}\parallel }_{\infty }^2ϵ.\hfill \end{array}$$

Hence, for each $f\in {\mathcal{F}}_{\alpha }$ with ${\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\le 1$, there is a ${\delta }_0\in (0,1)$ and ${\delta }_0={\delta }_0(f,ϵ)$ such that for $r\in ({\delta }_0,1)$,

$$\begin{array}{c}\hfill {\int }_{|\phi (z)|>r}|f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<ϵ.\end{array}$$

(17)

Since $C_{\phi ,g}^n$ is a compact operator, it maps bounded sets into totally bounded sets. Therefore, for every $ϵ>0$, there is a finite collection of functions $f_1,\cdots ,f_k$ with $\parallel f_j{\parallel }_{{\mathcal{F}}_{\alpha }}\le 1$, $1\le j\le k$, such that for each $f\in {\mathcal{F}}_{\alpha }$ with ${\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\le 1$, there is a $j\in \{1,2,\cdots ,k\}$ such that

$$\begin{array}{c}\hfill {\int }_{\mathbb{D}}|f^{\left(n\right)}\left(\phi \left(z\right)\right)-f_j^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<ϵ.\end{array}$$

(18)

Conversely, it follows from (17) that if $\delta ={max}_{1\le j\le k}{\delta }_j(f_j,ϵ)$, then for $r\in (\delta ,1)$ and all $j\in \{1,2,\cdots ,k\}$, we have

$$\begin{array}{c}\hfill {\int }_{|\phi (z)|>r}|f_j^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)<ϵ.\end{array}$$

(19)

Therefore, from (18) and (19), for $r\in (\delta ,1)$ and every $f\in {\mathcal{F}}_{\alpha }$ with ${\parallel f\parallel }_{{\mathcal{F}}_{\alpha }}\le 1$, we get

$$\begin{array}{c}\hfill {\int }_{|\phi (z)|>r}|f^{\left(n\right)}{\left(\phi \left(z\right)\right)|}^2{|g\left(z\right)|}^2\omega \left(z\right)dA\left(z\right)⪯ϵ.\end{array}$$

(20)

Now, consider the function $f\left(z\right)=f_{\xi }\left(z\right)={\displaystyle \frac{1}{{(1-\overline{\xi }z)}^{\alpha }}}$, $\xi \in \mathbb{T}$ and apply (20) to these functions; we obtain

$$\underset{\xi \in \mathbb{T}}{sup}{\int }_{|\phi (z)|>r}{\displaystyle \frac{{|g\left(z\right)|}^2}{|1-\overline{\xi }{\phi \left(z\right)|}^{2(n+\alpha )}}}\omega \left(z\right)dA\left(z\right)⪯ϵ.$$

The proof is complete.   □

Potential Applications

Compact operators are important not only for their well-developed theory but also because they arise naturally in many applications, such as integral equations in mathematical physics. The compactness criteria established in Corollary 2.6 and 3.4 can be used to determine when the integral equation

$${\int }_0^z{f}^{\left(n\right)}\left(\phi \left(\zeta \right)\right)g\left(\zeta \right)d\zeta =h\left(z\right)$$

has a solution in the weighted Bloch space ${\mathcal{B}}_{\omega }$ or the weighted Dirichlet space ${\mathcal{D}}_{\omega }$. Moreover, in the compact case, the operator $C_{\phi ,g}^n$ can be approximated by finite-rank operators, which allows the use of numerical methods. These results also apply to special cases of the operator, including composition operators, integral operators, and differentiation-composition operators, which have been widely studied in the literature.

4. Conclusions

In this paper, we systematically investigated the generalized integral-type operator $C_{\phi ,g}^n$ acting between the fractional Cauchy transform space ${\mathcal{F}}_{\alpha }$ and two important classes of analytic function spaces: the weighted Bloch space ${\mathcal{B}}_{\omega }$ and the weighted Dirichlet space ${\mathcal{D}}_{\omega }$. Our main objectives were to compute or estimate the operator and essential norms and characterize the boundedness and compactness of these operators.

For the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{B}}_{\omega }$, we established two equivalent exact formulas for the operator norm (see Theorems 1 and 2). These formulas are expressed in terms of the weight $\omega$, functions g and $\phi$, and parameters $\alpha$ and n, and they directly characterize boundedness (Corollary 1). Moreover, we obtained an estimate for the essential norm, showing that it is equivalent to a limit involving the weight and the symbol $\phi$ as $|\phi (z)|\to 1$ (Theorem 3). Consequently, we could derive a necessary and sufficient condition for the compactness of the operator (Corollary 2).

In the second part, we studied the operator $C_{\phi ,g}^n:{\mathcal{F}}_{\alpha }\to {\mathcal{D}}_{\omega }$. We provided an exact formula for the operator norm (Theorem 4) and, as a corollary, gave a boundedness criterion (Corollary 3). Furthermore, we characterized the compactness of this operator as a limiting condition on the integral kernel over the region where $|\phi (z)|$ is close to one (Theorem 5). The proof techniques applied here involved the use of test functions, measure representations for functions in ${\mathcal{F}}_{\alpha }$, and standard compactness criteria in Banach spaces of analytic functions.

The results presented here extend and unify several previous studies on composition operators, integral-type operators, and their generalizations on spaces of analytic functions. Exact norm formulas are obtained for both target spaces, and an essential norm estimate (in the sense of an equivalence) is provided for the Bloch case; for the Dirichlet case, a compactness criterion is established. Not only are these results of theoretical interest, but they also provide practical tools for verifying boundedness and compactness in concrete applications. Possible directions for future research include the study of similar operators on other function spaces, such as weighted Bergman or Hardy spaces, and the investigation of the essential norm for other operator classes.