Essential Norm of t-Generalized Composition Operators from F(p, q, s) to Iterated Weighted-Type Banach Space

Abstract

In this work, we characterize the boundedness of t-generalized composition operators from $F(p,q,s)$ spaces to iterated weighted-type Banach space. We also provide estimates of the norm and the essential norm of t-generalized composition operators from $F(p,q,s)$ spaces to iterated weighted-type Banach space. As corollaries, we obtain approximations of the essential norm of integral operators and generalized composition operators from $F(p,q,s)$ spaces to iterated weighted-type Banach space. Moreover, we conclude our work by discussing the t-generalized composition operators and the special cases of $F(p,q,s)$.

1. Introduction

Let $H\left(\mathbb{D}\right)$ denote the set of analytic functions on the open unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$, and let $S\left(\mathbb{D}\right)$ represent the set of analytic self-maps of $\mathbb{D}$.

For $\phi \in S\left(\mathbb{D}\right)$, the composition operator $C_{\phi }$ acting on $H\left(\mathbb{D}\right)$ is defined as follows:

$$\begin{array}{c}\hfill C_{\phi }f=f\circ \phi .\end{array}$$

(1)

In recent years, a growing focus has emerged on examining composition operators and their actions across various spaces of analytic functions. Particularly, significant attention has been devoted to exploring the intricate connections between $C_{\phi }$ and the properties of $\phi$. This area of research has been extensively investigated and discussed in works such as [1,2,3,4,5,6,7,8], along with the references cited therein.

Given $g\in H\left(\mathbb{D}\right)$, the integral operator $I_g$ is defined as

$$\begin{array}{c}\hfill \left(I_{g}f\right)\left(z\right)={\int }_0^z{f}^{\prime }\left(w\right)g\left(w\right)dw.\end{array}$$

(2)

Assuming that $g\in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$, a linear operator is defined as follows:

$$\begin{array}{c}\hfill \left(C_{\phi }^{g}f\right)\left(z\right)={\int }_0^z{f}^{\prime }\left(\phi \left(w\right)\right)g\left(w\right)dw.\end{array}$$

(3)

This operator is referred to as the generalized composition operator. If $\phi \left(z\right)=z$, $C_{\phi }^g$ reduces to the integral operator $I_g$. In the case where $g={\phi }^{\prime }$, it is observed that the operator $C_{\phi }^g$ becomes a composition operator since $C_{\phi }^{{\phi }^{\prime }}-C_{\phi }$ is constant. Thus, $C_{\phi }^g$ serves as a generalization of the composition operator introduced in [9].

The study of the boundedness and compactness of generalized composition operators on Bloch-type spaces and Zygmund spaces has been explored in [9]. In [10], a new characterization of the generalized composition operator on Zygmund spaces was presented. Additional insights into the generalized composition operator on various spaces can be found in related works such as [11,12,13,14].

Consider $g\in H\left(\mathbb{D}\right)$, $\phi \in S\left(\mathbb{D}\right)$, and $t\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$. Building upon the motivation provided by (1)–(3), Kamal, Abd-Elhafeez, and Eissa [15] introduced a new operator known as the t-generalized composition operator, defined as

$$\left(C_{\phi }^{g,t}f\right)\left(z\right)={\int }_0^z{f}^{\prime }\left(\phi \left(w\right)\right)g^{\left(t\right)}\left(w\right)dw.$$

This operator is an extension of the generalized composition operator. Specifically, when $t=0$, $C_{\phi }^{g,0}$ coincides with $C_{\phi }^g$. Unlike the generalized composition operator, the t-generalized composition operator accommodates varying degrees of differentiability, governed by the parameter t. This parameterization opens up new avenues for analyzing the interplay between operator properties and function space characteristics.

Let $\mu$ be a positive continuous function on $\mathbb{D}$, which we refer to as a weight, and $k\in {\mathbb{N}}_0$. In [16], Stević introduced the iterated weighted-type Banach space ${\mathcal{V}}_{\mu ,k}$ as follows:

$${\mathcal{V}}_{\mu ,k}=\left\{f\in H\left(\mathbb{D}\right):\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left|f^{\left(k\right)}\left(z\right)\right|<\infty \right\},$$

with the norm

$${\parallel f\parallel }_{{\mathcal{V}}_{\mu ,k}}:=\sum _{m=0}^{k-1}|f^{\left(m\right)}\left(0\right)|+\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left|f^{\left(k\right)}\left(z\right)\right|.$$

The little iterated weighted-type space ${\mathcal{V}}_{\mu ,k}^0$ is the closed subspace of ${\mathcal{V}}_{\mu ,k}$ such that

$$\underset{\left|z\right|\to 1}{lim}\mu \left(z\right)\left|f^{\left(k\right)}\left(z\right)\right|=0.$$

For $k=0,1,2$, the space ${\mathcal{V}}_{\mu ,k}$ is the weighted-type space $H_{\mu }^{\infty }$, the weighted Bloch-type space $B_{\mu }$, and the weighted Zygmund-type space $Z_{\mu }$, respectively.

Consider $\alpha >0$ and $\mu \left(z\right)={(1-|z|}^2{)}^{\alpha }$. When $n=1,2$, ${\mathcal{V}}_{\mu ,k}$ coincides with the Bloch-type space $B_{\alpha }$ and the Zygmund-type space $Z_{\alpha }$, respectively. In particular, for $\alpha =1$, we obtain the classical Bloch space B and the Zygmund space Z, respectively. Moreover, when $\mu \left(z\right)=1-{\left|z\right|}^2$, as proven in Theorem 1 of [17], ${\mathcal{V}}_{\mu ,k}$ serves as the dual of the Hardy space $H^{\frac{1}{k}}$ for all $k\ge 2$. For further details on these spaces, please refer to [18,19].

The iterated weighted-type Banach spaces have a significant role in the field of approximation theory and numerical analysis. They are particularly useful for measuring the precision of different numerical methods used to approximate functions with nth-order derivatives, like finite difference and finite element methods. Additionally, these spaces can be employed to determine the rates at which various approximation schemes converge and to calculate error limits for numerical solutions of differential equations. Additionally, they have applications in machine learning, where they are used to model complex data structures and make predictions based on them. More details can be found in [20,21,22].

Let $p>0$, $s\ge 0$, and $q>-2$ such that $q+s>-1$. The general family space $F(p,q,s)$ is the set of all analytic functions that satisfy

$${\parallel f\parallel }_{F(p,q,s)}:=\left|f\left(0\right)\right|+{\left(\underset{a\in \mathbb{D}}{sup}{\int }_{\mathbb{D}}|f^{\prime }{\left(w\right)|}^p{(1-|w|}^2{)}^q(1-|{\alpha }_a\left(w\right){{|}^2)}^{s}dm\left(w\right)\right)}^{1/p}<\infty ,$$

where $dm$ denotes the Lebesgue area measure such that $m\left(\mathbb{D}\right)=1$, and

$${\alpha }_a\left(z\right)=\frac{a-z}{1-\overline{a}z}.$$

The little space $F_0(p,q,s)$ is the closed subspace of $F(p,q,s)$ such that

$$\underset{\left|a\right|\to 1}{lim}{\int }_{\mathbb{D}}|f^{\prime }{\left(w\right)|}^p{(1-|w|}^2{)}^q(1-|{\alpha }_a\left(w\right){{|}^2)}^{s}dm\left(w\right)=0.$$

These spaces were introduced by Zhao [23]. Equipped with the above norm, the general family space $F(p,q,s)$ becomes a Banach space. It is well known in [24] that there is a positve constant C such that

$$\begin{array}{c}\hfill {(1-|z|}^2{)}^{m-1+\frac{q+2}{p}}|f^{\left(m\right)}{\left(z\right)|\le C\parallel f\parallel }_{F(p,q,s)}\forall m\in \mathbb{N},f\in F(p,q,s).\end{array}$$

(4)

Previous research efforts have made significant strides in characterizing the boundedness and compactness properties of operators across a variety of function spaces, ranging from $F(p,q,s)$ to several iterated weighted-type Banach spaces. For instance, Yang, as detailed in [25], provided a characterization of the boundedness and compactness of weighted differentiation composition operators from the $F(p,q,s)$ space to $B_{\alpha }$. Similarly, Ye, in [26], examined the boundedness and compactness of the weighted composition operator from the general family space $F(p,q,s)$ to the logarithmic Bloch space ${\mathcal{B}}_{log}$. Another contribution by Yang, discussed in [24], focused on investigating the boundedness and compactness of composition operators from the general family space $F(p,q,s)$ space to ${\mathcal{V}}_{\mu ,k}$. Zhou and Chen, in their work [27], conducted a study on the weighted composition operator from the $F(p,q,s)$ space to $B_{\alpha }$ on the unit ball. Additionally, in [28,29], Stević engaged in discussions concerning the boundedness and compactness of integral operators between $F(p,q,s)$ spaces and Bloch-type spaces within the unit ball. These investigations contribute significantly to our understanding of the behavior of various operators on different function spaces, shedding light on the intricate interplay between operator-theoretic properties and function-space characteristics.

Expanding upon this existing body of literature, our research introduces a novel operator, the t-generalized composition operator. This operator extends the concept of generalized composition operators to a new level of generality and flexibility, offering insights into previously unexplored areas of operator theory. What sets t-generalized composition operators apart is their ability to capture and manipulate higher-order derivative information, providing a richer framework for analyzing the composition of functions. By incorporating tth-order derivatives of the function g into the composition process, t-generalized composition operators offer a more nuanced understanding of how compositions interact with the underlying function spaces. This additional degree of control over the composition process enables us to explore a broader range of phenomena and derive more refined results. In particular, our study investigates the boundedness and essential norm of t-generalized composition operators as they operate from $F(p,q,s)$ spaces to iterated type spaces, providing valuable contributions to the understanding of these operators’ behaviors in diverse function-space settings. Furthermore, we discuss the special cases of $F(p,q,s)$ and the operator $C_{\phi }^{g,t}$.

In this work, we will consistently use the symbol C to represent a positive constant that remains independent of the variables or parameters involved, although its value may vary with each instance. The notation $A⪯B$ indicates that there exists a positive constant c such that $cA\le B$. Furthermore, we employ the notation $A\asymp B$ to signify that there exist positive constants $c_1$ and $c_2$, with $c_1\le c_2$, such that $c_{1}A\le B\le c_{2}A$.

2. Boundedness

The main goal of this section is to characterize the boundedness of t-generalized compostion operators from $F(p,q,s)$ spaces to iterated weighted-type Banach spaces.

Lemma 1

(Lemma 4, [16]). Given $f,g\in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$, for $n\in \mathbb{N}$ and $z\in \mathbb{D}$,

$${\left(g(f\circ \phi )\right)}^{\left(n\right)}\left(z\right)=\sum _{\ell =0}^n{f}^{(\ell )}\left(\phi \left(z\right)\right)\sum _{j=\ell }^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)g^{(n-j)}\left(z\right)A_{j,\ell }({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(j-\ell +1)}\left(z\right)),$$

where

$$\begin{array}{c}\hfill A_{j,\ell }({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(j-\ell +1)}\left(z\right)):=\sum _{{\ell }_1,{\ell }_2,\dots ,{\ell }_j}\frac{j!}{{\ell }_1!{\ell }_2!\dots {\ell }_j!}\prod _{m=1}^j{\left(\frac{{\phi }^{\left(m\right)}\left(z\right)}{m!}\right)}^{{\ell }_m},\end{array}$$

and the sum is taken over all nonnegative integers ${\ell }_1,\dots ,{\ell }_j$ such that $\ell ={\ell }_1+\dots +{\ell }_j$, and ${\ell }_1+2{\ell }_2+\dots +j{\ell }_j=j$.

Then, for the t-generalized compostion operator case, we have

$$\begin{array}{cc}\hfill & {\left(\left(C_{\phi }^{g,t}\right)f\right)}^{\left(n\right)}\left(z\right)\hfill \\ \hfill & ={\left(g^{\left(t\right)}(f^{\prime }\circ \phi )\right)}^{(n-1)}\left(z\right)\hfill \\ \hfill & =\sum _{\ell =1}^n{f}^{(\ell )}\left(\phi \left(z\right)\right)\sum _{j=\ell -1}^{n-1}\left(\genfrac{}{}{0pt}{}{n-1}{j}\right)g^{(t+n-1-j)}\left(z\right)A_{j,\ell -1}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(j-\ell +2)}\left(z\right)).\hfill \end{array}$$

(5)

We set $k\in \mathbb{N}$ and $t\in {\mathbb{N}}_0$, as well as functions g and $\phi$. For $z\in \mathbb{D},\ell \in \{1,\dots ,k\}$, we define

$${{\mathbf{N}}_{\ell }}^t\left(z\right):=\left|\sum _{j=\ell -1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(z\right)A_{j,\ell -1}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(j-\ell +2)}\left(z\right))\right|.$$

Theorem 1.

We set $k\in \mathbb{N}$ and $t\in {\mathbb{N}}_0$ and let $g\in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. Then, the following statements are equivalent.

(a) $C_{\phi }^{g,t}:F(p,q,s)\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(b) M:=$\underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\sum }_{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\frac{q+2}{p}+\ell -1}}<\infty .$

Moreover, if $C_{\phi }^{g,t}$ is bounded, then

$$\begin{array}{ccc}\hfill \parallel C_{\phi }^{g,t}\parallel & \asymp & \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.\hfill \end{array}$$

Proof. 

(b) ⇒ (a) Let $f\in F(p,q,s)$ such that ${\parallel f\parallel }_{F(p,q,s)}\le 1$ and $z\in \mathbb{D}$. By (4) and (5), we have

$$\begin{array}{cc}\hfill & \mu \left(z\right)|{\left(C_{\phi }^{g,t}f\right)}^{\left(k\right)}\left(z\right)|\hfill \\ & \le \mu \left(z\right)\sum _{\ell =1}^k\left|f^{(\ell )}\left(\phi \left(z\right)\right)\right|\left|\sum _{j=\ell -1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(z\right)A_{j,\ell -1}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(j-\ell +2)}\left(z\right))\right|\hfill \\ & ⪯\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.\hfill \end{array}$$

(6)

Taking the supremum over all z in $\mathbb{D}$, we obtain

$$\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left|{\left(C_{\phi }^{g,t}f\right)}^{\left(k\right)}\left(z\right)\right|⪯\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.$$

(7)

Noting $\left(C_{\phi }^{g,t}f\right)\left(0\right)=0$ and again by (4), for each $m\in \{1,\dots ,k-1\}$, we have

$$\begin{array}{cc}\hfill & |{\left(C_{\phi }^{g,t}f\right)}^{\left(m\right)}\left(0\right)|\hfill \\ & \le \sum _{\ell =1}^m\left|f^{(\ell )}\left(\phi \left(0\right)\right)\right|\left|\sum _{j=\ell -1}^{m-1}\left(\genfrac{}{}{0pt}{}{m-1}{j}\right)g^{(t+m-1-j)}\left(0\right)A_{j,\ell -1}({\phi }^{\prime }\left(0\right),\dots ,{\phi }^{(j-\ell +2)}\left(0\right))\right|\hfill \\ & ⪯\sum _{\ell =1}^m\frac{{{\mathbf{N}}_{\ell }}^t\left(0\right)}{{(1-|\phi \left(0\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}\hfill \\ & ⪯\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.\hfill \end{array}$$

(8)

Combining (7) and (8), we obtain

$$\parallel C_{\phi }^{g,t}{f\parallel }_{{\mathcal{V}}_{\mu ,k}}⪯\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}$$

which proves that $C_{\phi }^{g,t}$ is bounded.

By taking the supremum over all f in the unit ball of $F(p,q,s)$, we obtain the upper estimate.

(a) ⇒ (b) Let $k\in \mathbb{N}$ and $w,a\in \mathbb{D}$. By [30] and Lemma 3 in [16], for each $l\in \{0,\dots ,k\}$, there exist unique real numbers $c_0,\dots ,c_k$ such that

$$\begin{array}{c}\hfill f_a\left(z\right):=\sum _{j=0}^k\frac{c_j{(1-|a|}^2{)}^{j+1}}{{(1-\overline{a}z)}^{j+\frac{q+2}{p}}},z\in \mathbb{D},\end{array}$$

(9)

which satisfies the conditions

$$\begin{array}{cc}\hfill f_a^{\left(l\right)}\left(a\right)& =\frac{{\overline{a}}^l}{{(1-|a|}^2{)}^{l-1+\frac{q+2}{p}}}\sum _{j=0}^k{c}_j\prod _{r=0}^{l-1}(j+r+\frac{q+2}{p})=\frac{{\overline{a}}^l}{{(1-|a|}^2{)}^{l-1+\frac{q+2}{p}}},\hfill \\ \hfill f_a^{\left(t\right)}\left(a\right)& =0,\mathrm{for}t\in \{0,\dots ,k\}\setminus \left\{l\right\}.\hfill \end{array}$$

Moreover, $L:=\underset{a\in \mathbb{D}}{sup}{\parallel f_a\parallel }_{F(p,q,s)}<\infty$.

Since $C_{\phi }^{g,t}$ is bounded, then by (5), we obtain

$$\begin{array}{cc}\hfill & \mu \left(w\right)\left|\sum _{\ell =1}^k{f}^{(\ell )}\left(\phi \left(w\right)\right)\sum _{j=\ell -1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(w\right)A_{j,\ell -1}({\phi }^{\prime }\left(w\right),\dots ,{\phi }^{(j-\ell +2)}\left(w\right))\right|\hfill \\ & =\mu \left(w\right)|{\left(C_{\phi }^{g,t}{f}_{\phi \left(w\right)}\right)}^{\left(k\right)}\left(w\right)|\hfill \\ & \le L\parallel C_{\phi }^{g,t}\parallel \hfill \end{array}$$

(10)

where for fixed $\ell =0,\dots ,k$ and $m=0,\dots ,k$,

$$\begin{array}{c}\hfill |f_{\phi \left(w\right)}^{\left(m\right)}\left(\phi \left(w\right)\right)|=\left\{\begin{array}{cc}\frac{{\left|\phi \left(w\right)\right|}^{\ell }}{{(1-|\phi \left(w\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}\hfill & \mathrm{for}m=\ell \hfill \\ 0\hfill & \mathrm{for}m\ne \ell .\hfill \end{array}\right.\end{array}$$

(11)

Hence, by (10), we obtain

$$\begin{array}{c}\hfill \frac{{\mu \left(w\right)\left|\phi \left(w\right)\right|}^{\ell }{{\mathbf{N}}_{\ell }}^t\left(w\right)}{{(1-|\phi \left(w\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}\le L\parallel C_{\phi }^{g,t}\parallel .\end{array}$$

Therefore, if $\left|\phi \right(w\left)\right|>1/2$, then

$$\begin{array}{c}\hfill \frac{\mu \left(w\right){{\mathbf{N}}_{\ell }}^t\left(w\right)}{{(1-|\phi \left(w\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}\le \frac{L}{{\left|\phi \left(w\right)\right|}^{\ell }}\parallel C_{\phi }^{g,t}\parallel \le 2^{\ell }L\parallel C_{\phi }^{g,t}\parallel .\end{array}$$

(12)

On the other hand, when $\left|\phi \right(w\left)\right|\le 1/2$, it follows that for each $\ell \in \{1,\dots ,k\}$, we have

$$\begin{array}{c}\hfill \frac{{{\mathbf{N}}_{\ell }}^t\left(w\right)}{{(1-|\phi \left(w\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}\le {\left(\frac{4}{3}\right)}^{\ell -1+\frac{q+2}{p}}{{\mathbf{N}}_{\ell }}^t\left(w\right).\end{array}$$

(13)

Combining (12) and (13), it follows that to prove that

$$\frac{\mu \left(w\right){{\mathbf{N}}_{\ell }}^t\left(w\right)}{{(1-|\phi \left(w\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}\le C\parallel C_{\phi }^{g,t}\parallel ,$$

(14)

and it suffices to show that

$$\begin{array}{c}\hfill \mu \left(w\right){{\mathbf{N}}_{\ell }}^t\left(w\right)\le C\parallel C_{\phi }^{g,t}\parallel .\end{array}$$

(15)

For a non-negative integer n, let $p_n\left(z\right)=z^n$. By Proposition 2.13 in [23], $p_n\in F(p,q,s)$. Moreover, for all $n\in \{0,\dots ,k\}$, $\parallel p_n{\parallel }_{F(p,q,s)}$ is bounded by a constant C.

We establish (15) using an induction proof on $\ell \in \{1,\dots ,k\}$. For $\ell =1$, we have

$$\begin{array}{cc}\hfill & \mu \left(w\right)|{\left(\left(C_{\phi }^{g,t}\right)p_1^{(\ell )}\phi \left(w\right)\right)}^{\left(k\right)}\left(w\right)|\hfill \\ \hfill & =\mu \left(w\right)\left|\sum _{\ell =1}^k{p}_1^{(\ell )}\left(\phi \left(w\right)\right)\sum _{j=\ell -1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(w\right)A_{j,\ell -1}({\phi }^{\prime }\left(w\right),\dots ,{\phi }^{(j-\ell +2)}\left(w\right))\right|\hfill \\ & =\mu \left(w\right){\mathbf{N}}_1^t\left(w\right)\hfill \\ & \le C\parallel C_{\phi }^{g,t}\parallel .\hfill \end{array}$$

Therefore, we have

$$\mu \left(w\right){\mathbf{N}}_1^t\left(w\right)\le C\parallel C_{\phi }^{g,t}\parallel .$$

Assume that for $n\in \{1,...,\ell -1\}$, we have

$$\begin{array}{c}\hfill \mu \left(w\right){\mathbf{N}}_n^t\left(w\right)\le C\parallel C_{\phi }^{g,t}\parallel .\end{array}$$

Observe that

$$p_{\ell }^{\left(j\right)}\left(z\right)=\left\{\begin{array}{cc}\ell \dots (\ell -j+1)z^{\ell -j}\hfill & \mathrm{for}j=0,\dots \ell \hfill \\ 0\hfill & \mathrm{for}j=\ell +1,\dots ,k.\hfill \end{array}\right.$$

Therefore, we have

$$\begin{array}{ccc}\hfill & C\parallel C_{\phi }^{g,t}\parallel \hfill & \\ \hfill & \ge \mu \left(w\right)|{\left(C_{\phi }^{g,t}{p}_{\ell }\right)}^{\left(k\right)}\left(w\right)|\hfill \\ \hfill & =\mu \left(w\right)\left|\sum _{n=1}^k{p}_{\ell }^{\left(n\right)}\left(\phi \left(w\right)\right)\sum _{j=n-1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(z\right)A_{j,n-1}({\phi }^{\prime }\left(w\right),\dots ,{\phi }^{(j-n+2)}\left(w\right))\right|\hfill \\ & =\mu \left(w\right)|\sum _{n=1}^{\ell }{p}_{\ell }^{\left(n\right)}\left(\phi \left(w\right)\right)\sum _{j=n-1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(z\right)A_{j,n-1}({\phi }^{\prime }\left(w\right),\dots ,{\phi }^{(j-n+2)}\left(w\right))\hfill \\ & +\sum _{n=\ell +1}^k{p}_{\ell }^{\left(n\right)}\left(\phi \left(w\right)\right)\sum _{j=n-1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(z\right)A_{j,n-1}({\phi }^{\prime }\left(w\right),\dots ,{\phi }^{(j-n+2)}\left(w\right))|\hfill \\ & =\mu \left(w\right)|\sum _{n=1}^{\ell -1}\ell \dots (\ell -n+1){\left(\phi \left(w\right)\right)}^{\ell -n}\hfill \\ & \times \sum _{j=n-1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(z\right)A_{j,n-1}({\phi }^{\prime }\left(w\right),\dots ,{\phi }^{(j-n+2)}\left(w\right))\hfill \\ & +\ell !\sum _{j=\ell -1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(t+k-1-j)}\left(z\right)A_{j,n-1}({\phi }^{\prime }\left(w\right),\dots ,{\phi }^{(j-\ell +2)}\left(w\right))|.\hfill \end{array}$$

Therefore, we have

$$\begin{array}{cc}\hfill \ell !\mu \left(w\right){{\mathbf{N}}_{\ell }}^t& \le C\parallel C_{\phi }^{g,t}\parallel +\mu \left(w\right)\sum _{n=1}^{\ell -1}\ell \dots (\ell -n+1){\mathbf{N}}_n⪯\parallel C_{\phi }^{g,t}\parallel .\hfill \end{array}$$

By (12) and (14), for each $w\in \mathbb{D}$, we obtain

$$\frac{\mu \left(w\right){{\mathbf{N}}_{\ell }}^t\left(w\right)}{{(1-|\phi \left(w\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}⪯\parallel C_{\phi }^{g,t}\parallel .$$

(16)

By summing over all $\ell \in \{1,\dots ,k\}$ and taking the supremum over all w in $\mathbb{D}$, we obtain

$$\underset{w\in \mathbb{D}}{sup}\mu \left(w\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(w\right)}{{(1-|\phi \left(w\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}⪯\parallel C_{\phi }^{g,t}\parallel ,$$

which completes our proof. □

Focusing on the component operators $C_{\phi }^g$ and $I_g$, we derive the following two results.

Corollary 1.

Let $k\in \mathbb{N}$, $g\in H\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then, the following statements are equivalent.

(a) $C_{\phi }^g:F(p,q,s)\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(b) $\underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\sum }_{\ell =1}^k\frac{|{\sum }_{j=\ell -1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(k-1-j)}\left(z\right)A_{j,\ell -1}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(j-\ell +2)}\left(z\right))|}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}<\infty .$

Moreover, if $C_{\phi }^g$ is bounded, then

$$\begin{array}{ccc}\hfill \parallel C_{\phi }^g\parallel & \asymp & \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{|{\sum }_{j=\ell -1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(k-1-j)}\left(z\right)A_{j,\ell -1}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(j-\ell +2)}\left(z\right))|}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.\hfill \end{array}$$

Corollary 2.

Let $k\in \mathbb{N}$, and let $g\in H\left(\mathbb{D}\right)$. Then, the following statements are equivalent.

(a) $I_g:F(p,q,s)\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(b) $\underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\sum }_{\ell =1}^k\frac{|g^{(k-\ell -2)}\left(z\right)|}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}<\infty .$

Moreover, if $I_g$ is bounded, then

$$\begin{array}{ccc}\hfill \parallel I_g\parallel & \asymp & \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{|g^{(k-\ell )}\left(z\right)|}{{(1-|z|}^2{)}^{\ell -1+\frac{q+2}{p}}}.\hfill \end{array}$$

3. Essential Norm

The result presented in [7] is crucial for characterizing the compactness of the operators under investigation in this study.

Lemma 2

([7], Lemma 3.7). Let $X,Y$ be Banach spaces of analytic functions on $\mathbb{D}$, and let $T:X\to Y$ be a bounded linear operator. Suppose the following:

(i)

The point evaluation functionals on X are continuous;

(ii)

The closed unit ball of X is a compact subset of X in the topology of uniform convergence on compact sets;

(iii)

T is continuous when X and Y are given the topology of uniform convergence on compact sets.

Then, T is a compact operator if and only if for any bounded sequence $\left\{f_n\right\}$ in X such that $f_n$ converges uniformly to zero on compact sets, the sequence $\left\{T{f}_n\right\}$ converges to zero in the norm of Y.

Recall that the essential norm of a bounded linear operator $W:X\to Y$, where X and Y are Banach spaces, is given by

$${\parallel W\parallel }_e:=inf\left\{\parallel W-T\parallel /T:X\to Y\mathrm{compact}\right\}.$$

Therefore, a bounded linear operator W is compact if and only ${\parallel W\parallel }_e=0$.

The following lemma will be used to prove the main result of this section, and the proof is similar to the one in Lemma 3.1 in [31].

Lemma 3.

Let $k\in \mathbb{N}$, and let $0\le r<1$. For $f\in F(p,q,s)$, the dilation function $W_r$ in $F(p,q,s)$ is defined by $W_{r}f\left(z\right):=f\left(rz\right)$ for all $z\in \mathbb{D}$. Then, $W_r$ is compact on $F(p,q,s)$ and

$$\begin{array}{c}\hfill \tau :=\underset{0\le r<1}{sup}\parallel W_r\parallel <\infty .\end{array}$$

(17)

Moreover, for $\epsilon >0$ and $a\in (0,1)$, there exists $r\in (0,1)$ such that

$$\begin{array}{c}\hfill \underset{{\parallel f\parallel }_{F(p,q,s)}=1}{sup}\underset{\left|z\right|\le a}{sup}|\left((I-W_r)f\right){)}^{\left(j\right)}\left(z\right)|<\epsilon ,\mathit{for}\mathit{all}j=1,\dots ,k.\end{array}$$

(18)

Now, we are ready to state the main result of this section.

Theorem 2.

Let $k\in \mathbb{N}$, $t\in {\mathbb{N}}_0$, $g\in H\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. If $C_{\phi }^{g,t}:F(p,q,s)\to {\mathcal{V}}_{\mu ,k}$ is bounded, then

$$\parallel C_{\phi }^{g,t}{\parallel }_e\asymp \underset{a\to 1}{lim}\underset{\left|\phi \right(z\left)\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.$$

Proof. 

To prove the upper estimate, let $a\in (0,1)$, $\epsilon >0$, and $0\le r<1$. $C_{\phi }^{g,t}{W}_r$ is compact, since $W_r$ is compact and $C_{\phi }^{g,t}$ is bounded. Then, by (4), (6), (17), and (18), we have the following:

$$\begin{array}{ccc}\hfill \parallel C_{\phi }^{g,t}{\parallel }_e& \le & \parallel C_{\phi }^{g,t}-C_{\phi }^{g,t}{W}_r\parallel \hfill \\ & =& \underset{{\parallel f\parallel }_{F(p,q,s)}=1}{sup}{\parallel \left(C_{\phi }^{g,t}(I-W_r)\right)f\parallel }_{{\mathcal{V}}_{\mu ,k}}\hfill \\ & =& \underset{{\parallel f\parallel }_{F(p,q,s)}=1}{sup}\left(\sum _{j=1}^{k-1}|{\left(C_{\phi }^{g,t}(I-W_r)f\right)}^{\left(j\right)}\left(0\right)|+\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)|{\left(C_{\phi }^{g,t}(I-W_r)f\right)}^{\left(k\right)}\left(z\right)|\right)\hfill \\ & \le & (k-1)\epsilon +\underset{{\parallel f\parallel }_{F(p,q,s)}=1}{sup}\underset{|\phi (z)|\le a}{sup}\mu \left(z\right)|{\left(C_{\phi }^{g,t}(I-W_r)f\right)}^{\left(k\right)}\left(z\right)|\hfill \\ \hfill & & +\underset{{\parallel f\parallel }_{F(p,q,s)}=1}{sup}\underset{a<|\phi (z)|<1}{sup}\mu \left(z\right)|{\left(C_{\phi }^{g,t}(I-W_r)f\right)}^{\left(k\right)}\left(z\right)|\hfill \\ & \le & (k-1)\epsilon +\underset{{\parallel f\parallel }_{F(p,q,s)}=1}{sup}\underset{|\phi (z)|\le a}{sup}\mu \left(z\right)\sum _{\ell =1}^k|{\left((I-W_r)f\right)}^{(\ell )}\left(\phi \left(z\right)\right)|{{\mathbf{N}}_{\ell }}^t\left(z\right)\hfill \\ & & +\underset{{\parallel f\parallel }_{F(p,q,s)}=1}{sup}\underset{a<|\phi (z)|<1}{sup}\mu \left(z\right)\sum _{\ell =1}^k|{\left((I-W_r)f\right)}^{(\ell )}\left(\phi \left(z\right)\right)|{{\mathbf{N}}_{\ell }}^t\left(z\right).\hfill \\ & \le & (k-1)\epsilon +\epsilon \underset{|\phi (z)|\le a}{sup}\mu \left(z\right)\sum _{\ell =1}^k{{\mathbf{N}}_{\ell }}^t\left(z\right)\hfill \\ & & +C\underset{{\parallel f\parallel }_{F(p,q,s)}=1}{sup}\underset{a<|\phi (z)|<1}{sup}\mu \left(z\right)\left[{\parallel f\parallel }_{F(p,q,s)}+{\parallel W_{r}f\parallel }_{F(p,q,s)}\right]\hfill \\ & & \times \sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}\hfill \\ & \le & (k-1+M)\epsilon +C(1+\tau )\underset{a<|\phi (z)|<1}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.\hfill \end{array}$$

For sufficiently small $\epsilon$, we obtain

$$\parallel C_{\phi }^{g,t}{\parallel }_e⪯\underset{a\to 1}{lim}\underset{\left|\phi \right(z\left)\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.$$

To prove the lower estimate, let $\left\{w_n\right\}$ be a sequence in $\mathbb{D}$ such that $\left|\phi \right(w_n\left)\right|\to 1$ and let $\ell \in \{1,\dots ,k\}$. Then, the sequence $f_n:=f_{\phi \left(w_n\right)}$ defined in the proof of Theorem 1 converges to 0 uniformly on compact subsets. Moreover, $G:=\underset{n\in \mathbb{N}}{sup}{\parallel f_n\parallel }_{F(p,q,s)}<\infty$.

Let $W:F(p,q,s)\to {\mathcal{V}}_{\mu ,k}$ be a compact operator. Then, by Lemma 2, $\underset{n\to \infty }{lim}{\parallel W{f}_n\parallel }_{{\mathcal{V}}_{\mu ,k}}=0$. Hence, by (5) and (11), we have

$$\begin{array}{cc}\hfill G\parallel C_{\phi }^{g,t}-W\parallel & \ge \underset{n\to \infty }{lim\; sup}{\parallel (C_{\phi }^{g,t}-W)f_n\parallel }_{{\mathcal{V}}_{\mu ,k}}\hfill \\ & \ge \underset{n\to \infty }{lim\; sup}\mu \left(w_n\right)\left|{\left(C_{\phi }^{g,t}{f}_n\right)}^{\left(k\right)}\left(w_n\right)\right|\hfill \\ & =\underset{n\to \infty }{lim\; sup}\frac{\mu \left(w_n\right){{\mathbf{N}}_{\ell }}^t\left(w_n\right)}{(1-|\phi \left(w_n\right){{|}^2)}^{\ell -1+\frac{q+2}{p}}}.\hfill \end{array}$$

Summing over all $\ell \in \{1,\dots ,k\}$ and taking the infimum over all compact operators $W:F(p,q,s)\to {\mathcal{V}}_{\mu ,k}$, we obtain

$$\underset{n\to \infty }{lim\; sup}\mu \left(w_n\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(w_n\right)}{(1-|\phi \left(w_n\right){{|}^2)}^{\ell -1+\frac{q+2}{p}}}⪯{\parallel C_{\phi }^{g,t}\parallel }_e.$$

□

Focusing on the component operators $C_{\phi }^g$ and $I_g$, we derive the following results.

Corollary 3.

Let $k\in \mathbb{N}$, $g\in H\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. If $C_{\phi }^g:F(p,q,s)\to {\mathcal{V}}_{\mu ,k}$ is bounded, then

$$\parallel C_{\phi }^g{\parallel }_e\asymp \underset{a\to 1}{lim}\underset{\left|\phi \right(z\left)\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{\left|{\sum }_{j=\ell -1}^{k-1}\left(\genfrac{}{}{0pt}{}{k-1}{j}\right)g^{(k-1-j)}\left(z\right)A_{j,\ell -1}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(j-\ell +2)}\left(z\right))\right|}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.$$

Corollary 4.

Let $k\in \mathbb{N}$ and $\phi \in S\left(\mathbb{D}\right)$. If $I_g:F(p,q,s)\to {\mathcal{V}}_{\mu ,k}$ is bounded, then

$$\parallel I_g{\parallel }_e\asymp \underset{a\to 1}{lim}\underset{\left|z\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{|g^{(k-\ell )}\left(z\right)|}{{(1-|z|}^2{)}^{\ell -1+\frac{q+2}{p}}}.$$

4. The Special Cases of the Space of $\mathit{F}(\mathit{p},\mathit{q},\mathit{s})$ and the Operators ${\mathit{C}}_{\mathit{\phi }}^{\mathit{g},\mathit{t}}$

We conclude this paper by exploring several special cases of $F(p,q,s)$ and $C_{\phi }^{g,t}$. To accomplish this, we begin by stating some fundamental definitions.

The space BMOA of analytic functions of bounded mean oscillation, defined as the space of analytic functions on unit disk such that

$${\parallel f\parallel }_{*}=\underset{a\in \mathbb{D}}{sup}{\parallel f\circ {\alpha }_a-f\left(a\right)\parallel }_{H^2},$$

where $H^2$ is the Hilbert Hardy space. With the norm

$${\parallel f\parallel }_{BMOA}:=\left|f\left(0\right)\right|+{\parallel f\parallel }_{*},$$

BMOA is a Banach space.

For $q>-1$, the weighted Dirichlet $D_q$ is the collection of all analytic functions $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ on $\mathbb{D}$ such that

$$\sum _{n=0}^{\infty }{n}^{1-q}{\left|a_n\right|}^2<\infty .$$

For $p\ge 1$, the Bergman space $L_a^p$ is defined as the space of all functions $f\in H\left(\mathbb{D}\right)$ such that

$${\int }_{\mathbb{D}}{\left|f\left(z\right)\right|}^{p}dA\left(z\right)<\infty .$$

$L_a^p$ is a Banach space with the norm

$${\parallel f\parallel }_{L_a^p}:={\left({\int }_{\mathbb{D}}{\left|f\left(z\right)\right|}^{p}dA\left(z\right)\right)}^{1/p}<\infty .$$

For $p>1$, an analytic function f on $\mathbb{D}$ belongs to Besov space $B^p$ if

$${\parallel f\parallel }_{B^p}=\left|f\left(0\right)\right|+{\left({\int }_{\mathbb{D}}{\left|f^{\prime }\left(z\right)\right|}^2{\left(1-{\left|z\right|}^2\right)}^{p-2}dA\left(z\right)\right)}^{\frac{1}{p}}<\infty .$$

In [23], Zhao proved that the above spaces coincide with $F(p,q,s)$ as follows:

• $F(p,q,s)=B_{\frac{q+2}{p}}$ for $s>1$;
• $F(p,p-2,s)=B$ for $s>1$;
• $F(2,1,0)=H^2$;
• $F(2,0,1)=BMOA$;
• $F(p,p,0)=L_a^p$ for $p\ge 1$;
• $F(p,p-2,0)=B^p$ for $p>1$;
• $F(2,q,0)=D_q$ for $q>-1$.

Therefore, using Theroems 1 and 2, we deduce the following:

Corollary 5.

Let $k\in \mathbb{N}$, $t\in {\mathbb{N}}_0$, $p>1$, $g\in H\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then, the following statements are equivalent.

(a) $C_{\phi }^{g,t}:B^p\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(b) $C_{\phi }^{g,t}:BMOA\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(c) $C_{\phi }^{g,t}:B\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(d) $\underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\sum }_{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell }}<\infty .$

Moreover, if $C_{\phi }^{g,t}$ is bounded, then

$$\begin{array}{cc}\hfill \parallel C_{\phi }^{g,t}\parallel & \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell }},\hfill \\ \hfill \parallel C_{\phi }^{g,t}{\parallel }_e& \asymp \underset{a\to 1}{lim}\underset{\left|\phi \right(z\left)\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell }}.\hfill \end{array}$$

Corollary 6.

Let $k\in \mathbb{N}$, $t\in {\mathbb{N}}_0$, $p>0$, and $q>-2$. Let $g\in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. Then, the following statements are equivalent.

(a) $C_{\phi }^{g,t}:B_{\frac{q+2}{p}}\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(b) $\underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\sum }_{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}<\infty .$

Moreover, if $C_{\phi }^{g,t}$ is bounded, then

$$\begin{array}{cc}\hfill \parallel C_{\phi }^{g,t}\parallel & \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}},\hfill \\ \hfill \parallel C_{\phi }^{g,t}{\parallel }_e& \asymp \underset{a\to 1}{lim}\underset{\left|\phi \right(z\left)\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell -1+\frac{q+2}{p}}}.\hfill \end{array}$$

Corollary 7.

Let $k\in \mathbb{N}$, $t\in {\mathbb{N}}_0$, $g\in H\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then, the following statements are equivalent.

(a) $C_{\phi }^{g,t}:H^2\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(b) $\underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\sum }_{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{1}{2}}}<\infty .$

Moreover, if $C_{\phi }^{g,t}$ is bounded, then

$$\begin{array}{cc}\hfill \parallel C_{\phi }^{g,t}\parallel & \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{1}{2}}},\hfill \\ \hfill \parallel C_{\phi }^{g,t}{\parallel }_e& \asymp \underset{a\to 1}{lim}\underset{\left|\phi \right(z\left)\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{1}{2}}}.\hfill \end{array}$$

Corollary 8.

Let $k\in \mathbb{N}$, $t\in {\mathbb{N}}_0$, and $q>-1$. Let $g\in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. Then, the following statements are equivalent.

(a) $C_{\phi }^{g,t}:D_q\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(b) $\underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\sum }_{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{q}{2}}}<\infty .$

Moreover, if $C_{\phi }^{g,t}$ is bounded, then

$$\begin{array}{cc}\hfill \parallel C_{\phi }^{g,t}\parallel & \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{q}{2}}},\hfill \\ \hfill \parallel C_{\phi }^{g,t}{\parallel }_e& \asymp \underset{a\to 1}{lim}\underset{\left|\phi \right(z\left)\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{q}{2}}}.\hfill \end{array}$$

Corollary 9.

Let $k\in \mathbb{N}$, $t\in {\mathbb{N}}_0$, and $p\ge 1$. Let $g\in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. Then, the following statements are equivalent.

(a) $C_{\phi }^{g,t}:L_a^p\to {\mathcal{V}}_{\mu ,k}$ is bounded.

(b) $\underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\sum }_{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{2}{p}}}<\infty .$

Moreover, if $C_{\phi }^{g,t}$ is bounded, then

$$\begin{array}{cc}\hfill \parallel C_{\phi }^{g,t}\parallel & \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{2}{p}}},\hfill \\ \hfill \parallel C_{\phi }^{g,t}{\parallel }_e& \asymp \underset{a\to 1}{lim}\underset{\left|\phi \right(z\left)\right|>a}{sup}\mu \left(z\right)\sum _{\ell =1}^k\frac{{{\mathbf{N}}_{\ell }}^t\left(z\right)}{{(1-|\phi \left(z\right)|}^2{)}^{\ell +\frac{2}{p}}}.\hfill \end{array}$$