Sums of Generalized Weighted Composition Operators Acting from Besov and BMOA Spaces to Bloch Spaces

Abstract

This paper studies a generalized class of linear operators acting on spaces of analytic functions, defined by $P_n^{\psi ,\phi }\left(f\right)\left(z\right)={\displaystyle \sum _{j=0}^n}{\psi }_j\left(z\right)f^{\left(j\right)}\left(\phi \left(z\right)\right)$, where $\psi =\{{\psi }_0,{\psi }_1,\dots ,{\psi }_n\}\subset H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. This formulation encompasses several classical operators, including composition, weighted composition, differentiation–composition, and the Stević–Sharma operator. We focus on the action of $P_n^{\psi ,\phi }$ from $BMOA$ and analytic Besov spaces ${\mathcal{B}}_p$ into the Bloch space $\mathcal{B}$, and provide necessary and sufficient conditions for boundedness and compactness. These results unify and extend many previously known characterizations and demonstrate the flexibility of the $P_n^{\psi ,\phi }$ framework in the context of analytic operator theory.

1. Introduction

Operator theory on spaces of analytic functions is a classical and active field at the intersection of complex analysis and functional analysis. It focuses on linear operators—often defined via composition, multiplication, or differentiation—that act as transformations on analytic functions and interact closely with the geometry of the domain. These operators capture deep structural properties of analytic function spaces and are central to the study of boundedness, compactness, and spectral behavior. As such, their analysis continues to be a major area of research in modern mathematical analysis.

We denote by $H\left(\mathbb{D}\right)$ the family of analytic functions on the open unit disk $\mathbb{D}$. The subspace $H^{\infty }$, consisting of bounded analytic functions, forms a Banach space with the norm

$${\parallel f\parallel }_{\infty }=\underset{z\in \mathbb{D}}{sup}\left|f\left(z\right)\right|.$$

The Bloch space $\mathcal{B}$ is defined by

$$\mathcal{B}=\left\{f\in H\left(\mathbb{D}\right):\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left)\right|f^{\prime }\left(z\right)|<\infty \right\},$$

and becomes a Banach space when equipped with the norm

$${\parallel f\parallel }_{\mathcal{B}}=\left|f\left(0\right)\right|+\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left)\right|f^{\prime }\left(z\right)|.$$

Its closed subspace, the little Bloch space ${\mathcal{B}}_0$, consists of functions satisfying

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

It is known [1] that ${\mathcal{B}}_0$ coincides with the closure in $\mathcal{B}$ of the set of polynomials, and therefore ${\mathcal{B}}_0$ is separable.

The Hardy space $H^2$ consists of all functions $f\in H\left(\mathbb{D}\right)$ such that

$${\parallel f\parallel }_{H^2}^2=\underset{0<r<1}{sup}{\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{\left|f\left(r{e}^{i\theta }\right)\right|}^{2}d\theta <\infty .$$

This space is a Hilbert space, with the inner product given by

$${\langle f,g\rangle }_{H^2}=\sum _{n=0}^{\infty }{a}_n\overline{b_n},\mathrm{for}f\left(z\right)=\sum a_n{z}^n,g\left(z\right)=\sum b_n{z}^n.$$

It is also known that $H^{\infty }\subset H^2\subset \mathcal{B}$ with continuous inclusions.

Another important space is $BMOA$, the space of analytic functions of bounded mean oscillation. A function $f\in H\left(\mathbb{D}\right)$ belongs to $BMOA$ provided that

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

where, for $a\in \mathbb{D}$,

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

The corresponding Banach norm is given by

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

Since $H^{\infty }\subset BMOA$, and for $f\in H^{\infty }$ one has

$$\parallel f\circ {\alpha }_a{-f\left(a\right)\parallel }_{H^2}^2={\int }_{\partial \mathbb{D}}|f\left({\alpha }_a\left(\xi \right)\right){|}^{2}dm\left(\xi \right)-{\left|f\left(a\right)\right|}^2\le {\parallel f\parallel }_{\infty }^2,$$

it follows that

$${\parallel f\parallel }_{*}\le {\parallel f\parallel }_{\infty }.$$

(1)

Here m denotes the normalized Lebesgue measure on $\partial \mathbb{D}$ with $m(\partial \mathbb{D})=1$. Moreover, $BMOA$ is properly contained in $\mathcal{B}$ and the inclusion map is continuous (see [2,3]). For $f\in H^{\infty }$ one has

$${\parallel f\parallel }_{\mathcal{B}}\le {\parallel f\parallel }_{BMOA}\le 2{\parallel f\parallel }_{\infty }.$$

(2)

For $1<p<\infty$, the analytic Besov space ${\mathcal{B}}_p$ consists of functions $f\in H\left(\mathbb{D}\right)$ such that

$$b_p\left(f\right)={\left({\int }_{\mathbb{D}}|f^{\prime }{\left(w\right)|}^p{(1-|w|}^2{)}^{p-2}dA\left(w\right)\right)}^{1/p}<\infty ,$$

where $dA$ denotes the normalized area measure on $\mathbb{D}$. The space is normed by

$${\parallel f\parallel }_{{\mathcal{B}}_p}=\left|f\left(0\right)\right|+b_p\left(f\right).$$

When $p=2$, ${\mathcal{B}}_2$ coincides with the classical Dirichlet space $\mathcal{D}$, where an equivalent norm is

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

A space X of analytic functions on $\mathbb{D}$ with seminorm ${\parallel ·\parallel }_{sX}$ is called Möbius-invariant if

$${\parallel f\circ \phi \parallel }_{sX}={\parallel f\parallel }_{sX}$$

whenever $f\in X$ and $\phi$ is an automorphism of $\mathbb{D}$. It is classical that ${\mathcal{B}}_p$, $\mathcal{D}$, $BMOA$, and $\mathcal{B}$ are Möbius-invariant; see [3,4]. Furthermore,

$${\mathcal{B}}_p\subset BMOA\subset \mathcal{B},$$

with the estimate

$${\parallel f\parallel }_{\mathcal{B}}\le {\parallel f\parallel }_{BMOA}\le {\parallel f\parallel }_{{\mathcal{B}}_p},f\in {\mathcal{B}}_p.$$

(3)

Let $S\left(\mathbb{D}\right)$ denote the set of analytic self-maps of the open unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $\phi \in S\left(\mathbb{D}\right)$, the classical composition operator $C_{\phi }$ acting on the space $H\left(\mathbb{D}\right)$ of analytic functions on $\mathbb{D}$ is defined by

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

Composition operators have been investigated in depth; see the monographs [5,6] for an extensive account.

A natural extension of the composition operator is obtained by adding a weight function. If $\psi \in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$, the weighted composition operator $W_{\psi ,\phi }$ is defined by

$$W_{\psi ,\phi }f=\psi (f\circ \phi ),f\in H\left(\mathbb{D}\right).$$

Weighted composition operators arise naturally by combining multiplication and composition within analytic function spaces, forming a unified and versatile class of operators. They play a fundamental role in the study of isometries on Banach spaces of holomorphic functions. In particular, Forelli [7] showed that all isometries of the Hardy spaces $H^p$, for $p\ne 2$, over the unit disk $\mathbb{D}$, are precisely weighted composition operators; see, for instance, [8,9,10,11,12,13,14,15,16,17].

An even broader generalization is obtained by incorporating differentiation into the composition framework. Let $n\in {\mathbb{N}}_0$ and denote by $D^n$ the nth derivative operator, defined by

$$D^{n}f=f^{\left(n\right)},f\in H\left(\mathbb{D}\right).$$

Given $\psi \in H\left(\mathbb{D}\right)$, $\phi \in S\left(\mathbb{D}\right)$, and $n\in {\mathbb{N}}_0$, one defines

$$D_{\psi ,\phi }^{n}f=\psi (f^{\left(n\right)}\circ \phi ),$$

which is called the weighted differentiation composition operator. For further developments on weighted differentiation composition operators, see [18,19,20,21,22,23,24,25,26,27].

A notable extension of this idea was introduced by Stević, Sharma, and Bhat [28,29], who studied the operator

$$S_{\chi ,\psi ,\phi }\left(f\right)\left(z\right)=\chi \left(z\right)f\left(\phi \left(z\right)\right)+\psi \left(z\right)f^{\prime }\left(\phi \left(z\right)\right),$$

where $\chi ,\psi \in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. This operator, now commonly referred to as the Stević–Sharma operator, can be expressed as the linear combination

$$S_{\chi ,\psi ,\phi }=W_{\chi ,\phi }+D_{\psi ,\phi }^1.$$

Let $n\in {\mathbb{N}}_0$ and $\psi =\{{\psi }_0,{\psi }_1,\dots ,{\psi }_n\}$ with each ${\psi }_j\in H\left(\mathbb{D}\right)$. Define

$$P_n^{\psi ,\phi }f=\sum _{j=0}^n{\psi }_j(f^{\left(j\right)}\circ \phi )=\sum _{j=0}^n{D}_{{\psi }_j,\phi }^{\left(j\right)}f.$$

This operator, known as the sums of generalized weighted composition operators, was introduced and analyzed by Stević [30]. It provides a unified framework that includes many classical operators as particular cases. For instance, $P_0^{\psi ,\phi }$ corresponds to the weighted composition operator $W_{{\psi }_0,\phi }$, and when ${\psi }_j\equiv 0$ for all $j>0$, the classical operator $C_{\phi }$ is recovered. Hence, $P_n^{\psi ,\phi }$ unifies composition, weighted composition, and differentiation–composition operators, as well as the Stević–Sharma operator, within a single analytical model.

Research on such operators has become increasingly active. Colonna and Li [31] investigated weighted composition operators from Besov spaces into Bloch spaces, providing necessary and sufficient conditions for boundedness and compactness. Later, Colonna and Hmidouch [32] extended these results to operators mapping from $BMOA$ and the Bloch space into weighted-type spaces, offering unified characterizations of both bounded and compact mappings. In a related line of work, Liu, Lou, and Sharma [33] studied the weighted differentiation composition operator and established boundedness and compactness criteria for mappings from $BMOA$ and the Bloch space into Bloch-type spaces. More recently, Zhu and Hu [34] investigated the sums of generalized weighted composition operators acting on the Bloch space, obtaining results that highlight the versatility of this operator family.

Building on recent contributions, this paper investigates the operator $P_n^{\psi ,\phi }$ acting from $BMOA$ and analytic Besov spaces into the Bloch space, aiming to establish unified criteria for boundedness and compactness that generalize and subsume many known results. In particular, our approach encompasses earlier theorems on classical weighted composition, differentiation–composition, and Stević–Sharma operators as corollaries. Section 2 addresses boundedness of sums of generalized weighted composition operators, while Section 3 focuses on compactness, providing characterizations and deriving specific cases. This unified treatment not only simplifies existing results but also reveals deeper structural connections among analytic operators previously studied in isolation.

Throughout this paper, constants are denoted by C; they are positive and not necessarily the same at each occurrence.

2. Boundedness

We begin this section by establishing conditions for the boundedness of sums of generalized weighted composition operators acting from $BMOA$ and Bloch spaces into the Bloch space. To this end, we employ the following lemma.

Lemma 1

([32], Lemma 3.2). Fix $n\in \mathbb{N}$ and $a\in \mathbb{D}$. Then for each $k\in \{1,\dots ,n\}$ there is a unique finite sequence ${\left\{c_j\right\}}_{j=1}^{n+1}$ of real numbers such that the function

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

(4)

satisfies the conditions $f_a\left(a\right)=0$, $f_a^{\left(t\right)}\left(a\right)={\displaystyle \frac{{\overline{a}}^k}{{(1-|a|}^2{)}^k}}{\delta }_{k,t}$ for $t\in \{1,\dots ,n\}$, where ${\delta }_{k,t}=1$ if $t=k$ and ${\delta }_{k,t}=0$ if $k\ne t$.

Assume ${\psi }_{-1}=0$, ${\psi }_{n+1}^{\prime }=0$ for simplification. Define

$$M_j=\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^j}},0\le j\le n+1.$$

Theorem 1.

Fix $n\in {\mathbb{N}}_0$, and let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi =\{{\psi }_0,\dots ,{\psi }_n\}\subset H\left(\mathbb{D}\right)$. Then the following are equivalent.

(a)

$P_n^{\psi ,\phi }:\mathcal{B}\to \mathcal{B}$ is bounded.

(b)

$P_n^{\psi ,\phi }:BMOA\to \mathcal{B}$ is bounded.

(c)

$\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left|\right)|{\psi }_0^{\prime }\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}<\infty$ and ${\displaystyle \sum _{j=1}^{n+1}}{M}_j<\infty .$

Proof. 

That (a) implies (b) follows directly from the fact that $BMOA$ is continuously embedded in $\mathcal{B}$.

(b)⇒ (c) Our first step is to establish that $M_k<\infty$ for all $k=0,\dots ,n+1.$ For $k=0$, by the boundedness of $P_n^{\psi ,\phi }$ and since 1 belongs to $BMOA$, then

$$\begin{array}{cc}\hfill M_0=\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|=\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\left(P_n^{\psi ,\phi }1\right)}^{\prime }|& \le {\parallel 1\parallel }_{BMOA}{\parallel P_n^{\psi ,\phi }\parallel }_{BMOA\to \mathcal{B}}\hfill \\ & ={\parallel P_n^{\psi ,\phi }\parallel }_{BMOA\to \mathcal{B}}\hfill \end{array}$$

(5)

Next, let $k\in 1,\dots ,n+1$. For $a\in \mathbb{D}$, take the test function $f_a$ as defined in Lemma 1. Then $f_a\in H^{\infty }$ and

$$\parallel f_a{\parallel }_{\infty }=\underset{z\in \mathbb{D}}{sup}\left|\sum _{j=1}^{n+1}{\displaystyle \frac{c_j{(1-|a|}^2{)}^j}{{(1-\overline{a}z)}^j}}\right|\le \underset{z\in \mathbb{D}}{sup}\sum _{j=1}^{n+1}{\displaystyle \frac{|c_j\left|\right(1-{\left|a\right|}^2{)}^j}{|1-\overline{a}{z|}^j}}\le \sum _{j=1}^{n+1}\left|c_j\right|2^j.$$

Therefore, by (1), $f_a\in BMOA$ and $G:=\underset{a\in \mathbb{D}}{sup}{\parallel f_a\parallel }_{BMOA}<\infty$. Hence, by the boundedness of $P_n^{\psi ,\phi }:BMOA\to \mathcal{B}$, we get

$${(1-|w|}^2\left)\right|{\left(P_n^{\psi ,\phi }{f}_{\phi \left(w\right)}\right)}^{\prime }\left(w\right)|\le G\parallel P_n^{\psi ,\phi }{\parallel }_{BMOA\to \mathcal{B}}.$$

(6)

On the other hand, by Lemma 1, we have

$$\begin{array}{cc}\hfill & {(1-|w|}^2\left)\right|{\left(P_n^{\psi ,\phi }{f}_{\phi \left(w\right)}\right)}^{\prime }\left(w\right)|\hfill \\ & ={(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)f_{\phi \left(w\right)}\left(\phi \left(w\right)\right)+\sum _{j=1}^{n+1}\left[{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)\right]f_{\phi \left(w\right)}^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ & ={\displaystyle \frac{{(1-|w|}^2{\left)\right|\phi \left(w\right)|}^k|{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^k}}\hfill \end{array}$$

(7)

where $f_{\phi \left(w\right)}\left(\phi \left(w\right)\right)=0$ and $|f_{\phi \left(w\right)}^{\left(t\right)}\left(\phi \left(w\right)\right)|={\displaystyle \frac{{\left|\phi \left(w\right)\right|}^k}{{(1-|\phi \left(w\right)|}^2{)}^k}}{\delta }_{t,k}$ for $t\in \{1,\dots ,n+1\}$. Hence, from Equation (7) and inequality (6), we obtain

$${\displaystyle \frac{{\left|\phi \left(w\right)\right|}^k{(1-|w|}^2\left)\right|{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^k}}\le G{\parallel P_n^{\psi ,\phi }\parallel }_{BMOA\to \mathcal{B}}.$$

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

$${\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^k}}\le {\displaystyle \frac{1}{{\left|\phi \left(w\right)\right|}^k}}G\parallel P_n^{\psi ,\phi }{\parallel }_{BMOA\to \mathcal{B}}\le 2^{k}G{\parallel P_n^{\psi ,\phi }\parallel }_{BMOA\to \mathcal{B}}.$$

Thus, by taking the supremium over $\left|\phi \right(w\left)\right|>1/2$, we get

$$\underset{\left|\phi \right(w\left)\right|>1/2}{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^k}}<\infty .$$

(8)

On the other hand, if $\left|\phi \right(w\left)\right|\le 1/2$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^k}}\le {\left({\displaystyle \frac{4}{3}}\right)}^k{(1-|w|}^2\left)\right|{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)|.\end{array}$$

It suffices to show that

$$\begin{array}{c}\hfill \underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2\left)\right|{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)|<\infty .\end{array}$$

(9)

For $k=0$, we have by (5)

$$\begin{array}{c}\hfill \underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)+{\psi }_{-1}\left(w\right){\phi }^{\prime }\left(w\right)|=\underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|<M_0<\infty .\end{array}$$

Assume inductively that the statement is valid for every $j\in \{0,\dots ,k-1\}$,

$$\underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|<\infty .$$

Consider, for $m\in {\mathbb{N}}_0$, $p_m\left(z\right):=z^m$. Then $p_m$ is bounded and $\parallel p_m{\parallel }_{\infty }=1$, using (1), we have $\parallel p_k{\parallel }_{BMOA}\le 1$. Therefore

$${(1-|w|}^2\left)\right|{\left(P_n^{\psi ,\phi }{p}_k\right)}^{\prime }\left(w\right)|\le \parallel p_m{\parallel }_{BMOA}\parallel P_n^{\psi ,\phi }{\parallel }_{BMOA\to \mathcal{B}}\le {\parallel P_n^{\psi ,\phi }\parallel }_{BMOA\to \mathcal{B}}.$$

Observe that

$$\begin{array}{cc}\hfill & {\left(P_n^{\psi ,\phi }{p}_k\right)}^{\prime }\left(w\right)\hfill \\ & ={\left(\phi \left(w\right)\right)}^k{\psi }_0^{\prime }\left(w\right)+\sum _{j=1}^{n+1}[{\psi }_j^{\prime }\left(w\right)+{\psi }_j\left(w\right){\phi }^{\prime }\left(w\right)]p_k^{\left(j\right)}\left(\phi \left(w\right)\right)\hfill \\ & ={\left(\phi \left(w\right)\right)}^k{\psi }_0^{\prime }\left(w\right)+\sum _{j=1}^{k-1}k(k-1)\cdots (k-j+1){\left(\phi \left(w\right)\right)}^{k-j}[{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)]\hfill \\ & +k![{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)].\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & k!({\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right))\hfill \\ & ={\left(P_n^{\psi ,\phi }{p}_k\right)}^{\prime }\left(w\right)-{\left(\phi \left(w\right)\right)}^k{\psi }_0^{\prime }\left(w\right)\hfill \\ & -\sum _{j=1}^{k-1}k(k-1)\cdots (k-j+1){\left(\phi \left(w\right)\right)}^{k-j}[{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)].\hfill \end{array}$$

Multiplying by $1-{\left|w\right|}^2$, using the triangle inequality, and taking the supremum over all $\left|\phi \right(w\left)\right|\le 1/2$, we obtain

$$\begin{array}{cc}\hfill & k!\underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2\left)\right|{\psi }_k^{\prime }\left(w\right)+{\psi }_{k-1}\left(w\right){\phi }^{\prime }\left(w\right)\left)\right|\hfill \\ & \le \parallel P_n^{\psi ,\phi }\parallel +\underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2{\left)\right|\phi \left(w\right)|}^k\left|{\psi }_0^{\prime }\left(w\right)\right|\hfill \\ & +\underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2)\sum _{j=1}^{k-1}k(k-1)\cdots (k-j+1){\left(\phi \left(w\right)\right)}^{k-j}|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|\hfill \\ & \le \parallel P_n^{\psi ,\phi }\parallel +M_0+\sum _{j=1}^{k-1}k(k-1)\cdots (k-j+1)\underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|\hfill \\ & <\infty .\hfill \end{array}$$

Hence, by the inductive hypothesis, (9) is established for k, which completes the induction. Combining (8) and (9), we obtain that $M_k$ is finite for $k=0,\dots ,n+1$.

Next we show that

$$\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}<\infty .$$

Fix $w\in \mathbb{D}$. If $\left|\phi \right(w\left)\right|\le 1/2$, then

$$\begin{array}{c}\hfill \underset{\left|\phi \right(w\left)\right|\le 1/2}{sup}{(1-|w|}^2\left)\right|{\psi }^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\le log{\displaystyle \frac{4e}{3}}\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left)\right|{\psi }^{\prime }\left(z\right)|=log{\displaystyle \frac{4e}{3}}{M}_0<\infty .\end{array}$$

(10)

To analyze the case $\left|\phi \right(w\left)\right|>1/2$, we consider the function

$$g_{\phi \left(w\right)}\left(z\right)={\displaystyle \frac{{log}^2\frac{e}{1-\overline{\phi \left(w\right)}z}}{log\frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}}.$$

(11)

By Ref. [35], $g_{\phi \left(w\right)}\in BMOA$ and $N:=\underset{\left|\phi \right(w\left)\right|>1/2}{sup}{\parallel g_{\phi \left(w\right)}\parallel }_{BMOA}<\infty$.

Moreover, a direct computation shows that for $j\in \mathbb{N}$

$${\displaystyle \frac{d^j}{d{z}^j}}{g}_{\phi \left(w\right)}\left(z\right)={\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{-1}\left[log{\displaystyle \frac{e}{1-\overline{\phi \left(w\right)}z}}+\sum _{t=1}^{j-1}{\displaystyle \frac{1}{t}}\right]{\displaystyle \frac{2{\overline{\phi \left(w\right)}}^j(j-1)!}{{(1-\overline{\phi \left(w\right)}z)}^j}}.$$

In particular,

$$g_{\phi \left(w\right)}^{\left(j\right)}\left(\phi \left(w\right)\right)=\left[1+{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{-1}\sum _{t=1}^{j-1}{\displaystyle \frac{1}{t}}\right]{\displaystyle \frac{2{\overline{\phi \left(w\right)}}^j(j-1)!}{{(1-|\phi \left(w\right)|}^2{)}^j}}.$$

Hence

$$\begin{array}{c}N\parallel P_n^{\psi ,\phi }{\parallel }_{BMOA\to \mathcal{B}}\hfill \\ \ge {(1-|w|}^2)|{\left(P_n^{\psi ,\phi }{g}_{\phi \left(w\right)}\left(w\right)\right)}^{\prime }|\hfill \\ =(1-{\left|w\right|}^2)|{\psi }_0^{\prime }\left(w\right)g\left(\phi \left(w\right)\right)+\sum _{j=1}^{n+1}[{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)]g^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ =(1-{\left|w\right|}^2)|{\psi }_0^{\prime }\left(w\right)log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\hfill \\ +\sum _{j=1}^{n+1}\left[\left(1+{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{-1}\sum _{t=1}^{j-1}{\displaystyle \frac{1}{t}}\right){\displaystyle \frac{2\phi {\left(w\right)}^j(j-1)![{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)]}{{(1-|\phi \left(w\right)|}^2{)}^j}}\right|.\hfill \end{array}$$

(12)

Therefore

$$\begin{array}{cc}\hfill & {(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\hfill \\ & \le N\parallel P_n^{\psi ,\phi }{\parallel }_{BMOA\to \mathcal{B}}\hfill \\ & +(1-{\left|w\right|}^2)\sum _{j=1}^{n+1}[\left(1+{\left(log{\displaystyle \frac{4e}{3}}\right)}^{-1}\sum _{t=1}^{j-1}{\displaystyle \frac{1}{t}}\right){\displaystyle \frac{2(j-1)!|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^j}}.\hfill \end{array}$$

Thus

$$\begin{array}{cc}& \underset{\left|\phi \right(w\left)\right|>1/2}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\hfill \\ \hfill & <N\parallel P_n^{\psi ,\phi }{\parallel }_{BMOA\to \mathcal{B}}+\sum _{j=1}^{n+1}[\left(1+{\left(log{\displaystyle \frac{4e}{3}}\right)}^{-1}\sum _{t=1}^{j-1}{\displaystyle \frac{1}{t}}\right)2(j-1)!M_j<\infty .\hfill \end{array}$$

(13)

Combining (10) and (13), we conclude that $\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}$ is finite.

(c)⇒ (a) Let $f\in \mathcal{B}$. Then, for $z\in \mathbb{D}$, we have

$$\left|f\left(\phi \left(z\right)\right)\right|\le log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}{\parallel f\parallel }_{\mathcal{B}}.$$

Moreover, by Theorem 5.5 in [36], for each $k\in \mathbb{N}$ there exists $C>0$ such that

$${(1-|\phi \left(z\right)|}^2{)}^k|f^{\left(k\right)}{\left(\phi \left(z\right)\right)|\le C\parallel f\parallel }_{\mathcal{B}}.$$

(14)

Thus, we obtain

$$\begin{array}{cc}\hfill \parallel P_n^{\psi ,\phi }{f\parallel }_B& =|P_n^{\psi ,\phi }f\left(0\right)|+\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\left(P_n^{\psi ,\phi }f\right)}^{\prime }\left(w\right)|\hfill \\ & =\left|\sum _{j=0}^n{\psi }_j\left(0\right)f^{\left(j\right)}\left(\phi \left(0\right)\right)\right|\hfill \\ & +\underset{w\in \mathbb{D}}{sup}(1-{\left|w\right|}^2)\left|{\psi }_0^{\prime }\left(w\right)f\left(\phi \left(w\right)\right)+\sum _{j=1}^{n+1}[{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)]f^{\left(j\right)}\left(\phi \left(w\right)\right)\right|\hfill \\ & \le |{\psi }_0\left(0\right)\left|\right|f\left(\phi \left(0\right)\right)|+\sum _{j=1}^n|{\psi }_j\left(0\right)|\left|f^{\left(j\right)}\left(\phi \left(0\right)\right)\right|+\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)\left|\right|f\left(\phi \left(w\right)\right)|\hfill \\ & +\sum _{j=1}^{n+1}\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)\left|\right|f^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ & \le {C\parallel f\parallel }_{\mathcal{B}}(\left|{\psi }_0\left(0\right)\right|log{\displaystyle \frac{e}{1-{\left|\phi \left(0\right)\right|}^2}}+\sum _{j=1}^n{\displaystyle \frac{|{\psi }_j\left(0\right)|}{{(1-|\phi \left(0\right)|}^2{)}^j}}\hfill \\ & +\underset{w\in \mathbb{D}}{sup}\left|{\psi }_0^{\prime }\left(w\right)\right|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}+\sum _{j=1}^{n+1}{M}_j).\hfill \end{array}$$

Therefore $P_n^{\psi ,\phi }:\mathcal{B}\to \mathcal{B}$ is bounded. □

Our next task is to characterize the boundedness of sums of generalized weighted composition operators mapping Besov spaces into the Bloch space. To this end, we employ the following lemma.

Lemma 2.

Consider $f_a$ as in Lemma 1. Then $f_a\in {\mathcal{B}}_p$. Moreover $\underset{a\in \mathbb{D}}{sup}\parallel f_a\parallel <\infty$.

Proof. 

Let ${\alpha }_a\left(z\right)={\displaystyle \frac{a-z}{1-\overline{a}z}}$ for $a\in \mathbb{D}$. Observe that

$${\displaystyle \frac{1-{\left|a\right|}^2}{1-\overline{a}z}}=1-\overline{a}{\alpha }_a\left(z\right).$$

Therefore

$$\begin{array}{cc}\hfill f_a\left(z\right)=\sum _{j=1}^{n+1}{\displaystyle \frac{c_j{(1-|a|}^2{)}^j}{{(1-\overline{a}z)}^j}}& =\sum _{j=1}^{n+1}{c}_j{(1-\overline{a}{\alpha }_a\left(z\right))}^j\hfill \\ \hfill & =\sum _{j=1}^{n+1}{c}_j\sum _{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){(-\overline{a})}^k{\left({\alpha }_a\left(z\right)\right)}^k\hfill \\ & =\sum _{j=1}^{n+1}{c}_j\sum _{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){(-\overline{a})}^k(p_k\circ {\alpha }_a).\hfill \end{array}$$

Therefore, since ${\mathcal{B}}_p$ is möbius-invariant, we have

$$\begin{array}{cc}\hfill {\left(b_p\left(f_a\right)\right)}^p& ={\int }_{\mathbb{D}}|f_a^{\prime }{\left(z\right)|}^p{(1-|z|}^2{)}^{p-2}dA\hfill \\ & ={\int }_{\mathbb{D}}|\sum _{j=1}^{n+1}{c}_j\sum _{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){(-\overline{a})}^k{(p_k\circ {\alpha }_a)}^{\prime }{|}^p{{(1-|z|}^2)}^{p-2}dA\hfill \\ & \le \sum _{j=1}^{n+1}|c_j|\sum _{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){\int }_{\mathbb{D}}|{(p_k\circ {\alpha }_a)}^{\prime }\left(z\right){|}^p{{(1-|z|}^2)}^{p-2}dA\hfill \\ & =\sum _{j=1}^{n+1}|c_j|\sum _{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){\int }_{\mathbb{D}}|{\left(p_k\right)}^{\prime }\left(z\right){|}^p{{(1-|z|}^2)}^{p-2}dA\hfill \\ & =\sum _{j=1}^{n+1}|c_j|\sum _{k=1}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right)k{\int }_{\mathbb{D}}{\left|z\right|}^{p(k-1)}{{(1-|z|}^2)}^{p-2}dA\hfill \\ & \le \sum _{j=1}^{n+1}\left|c_j\right|\sum _{k=1}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right)k\hfill \\ & =\sum _{j=1}^{n+1}\sum _{k=1}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right)k\left|c_j\right|.\hfill \end{array}$$

Hence

$${\parallel f_a\parallel }_{{\mathcal{B}}_p}=|f_a\left(0\right)|+b_p\left(f_a\right)\le \sum _{j=1}^{n+1}\left|c_j\right|+{\left[\sum _{j=1}^{n+1}\sum _{k=1}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right)k\left|c_j\right|\right]}^{1/p}<\infty .$$

Moreover, $\underset{a\in \mathbb{D}}{sup}{\parallel f_a\parallel }_{{\mathcal{B}}_p}<\infty$. □

Theorem 2.

Fix $n\in {\mathbb{N}}_0$, and let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi =\{{\psi }_0,\dots ,{\psi }_n\}\subset H\left(\mathbb{D}\right)$. Then the following are equivalent.

(a)

$P_n^{\psi ,\phi }:{\mathcal{B}}_p\to \mathcal{B}$ is bounded.

(b)

$\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left)\right|{\psi }_0^{\prime }\left(z\right)|{[log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}]}^{1-1/p}<\infty$ and ${\sum }_{j=1}^{n+1}{M}_j<\infty .$

Proof. 

(b)⇒ (a) Let $f\in {\mathcal{B}}_p$. Then, by Theorem 9 in [37], we have

$$|f\left(\phi \left(z\right)\right)|\le C{[log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}]}^{1-1/p}{\parallel f\parallel }_{{\mathcal{B}}_p}.$$

Furthermore, using the continuous embedding ${\mathcal{B}}_p\hookrightarrow \mathcal{B}$, Theorem 5.5 of [36] combined with (3) implies that for every $k\in \mathbb{N}$ there is a constant $C>0$ satisfying

$${{(1-|\phi \left(z\right)|}^2)}^k|f^{\left(k\right)}{\left(\phi \left(z\right)\right)|\le C\parallel f\parallel }_{\mathcal{B}}\le C{\parallel f\parallel }_{{\mathcal{B}}_p}.$$

(15)

Thus

$$\begin{array}{cc}\hfill \parallel P_n^{\psi ,\phi }{f\parallel }_B& =|P_n^{\psi ,\phi }f\left(0\right)|+\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\left(P_n^{\psi ,\phi }f\right)}^{\prime }\left(w\right)|\hfill \\ \hfill & =\left|\sum _{j=0}^n{\psi }_j\left(0\right)f^{\left(j\right)}\left(\phi \left(0\right)\right)\right|\hfill \\ \hfill & +\underset{w\in \mathbb{D}}{sup}(1-{\left|w\right|}^2)\left|{\psi }_0^{\prime }\left(w\right)f\left(\phi \left(w\right)\right)+\sum _{j=1}^{n+1}[{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)]f^{\left(j\right)}\left(\phi \left(w\right)\right)\right|\hfill \\ & \le |{\psi }_0\left(0\right)\left|\right|f\left(\phi \left(0\right)\right)|+\sum _{j=1}^n|{\psi }_j\left(0\right)|\left|f^{\left(j\right)}\left(\phi \left(0\right)\right)\right|+\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)\left|\right|f\left(\phi \left(w\right)\right)|\hfill \\ & +\sum _{j=1}^{n+1}\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)\left|\right|f^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ \hfill & \le {C\parallel f\parallel }_{\mathcal{B}}(\left|{\psi }_0\left(0\right)\right|{\left[log{\displaystyle \frac{e}{1-{\left|\phi \left(0\right)\right|}^2}}\right]}^{1-1/p}+\sum _{j=1}^n{\displaystyle \frac{|{\psi }_j\left(0\right)|}{{(1-|\phi \left(0\right)|}^2{)}^j}}\hfill \\ & +\underset{w\in \mathbb{D}}{sup}\left|{\psi }_0^{\prime }\left(w\right)\right|{\left[log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right]}^{1-1/p}+\sum _{j=1}^{n+1}{M}_j).\hfill \end{array}$$

Therefore $P_n^{\psi ,\phi }:{\mathcal{B}}_p\to \mathcal{B}$ is bounded.

(a)⇒ (b) Since, by Lemma 2, $\underset{a\in \mathbb{D}}{sup}{\parallel f_a\parallel }_{{\mathcal{B}}_p}<\infty$, and following the argument employed in the proof of Theorem 1, the proof of $M_k<\infty$ for all $k=0,\dots ,n+1,$ proceeds by induction on k.

Next we show that

$$\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|{\left[log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right]}^{1-1/p}<\infty .$$

Fix $a\in \mathbb{D}$ and consider the following test function

$$h_a\left(z\right)={\displaystyle \frac{log\frac{e}{1-\overline{a}z}}{{\left(log\frac{e}{1-{\left|a\right|}^2}\right)}^{1/p}}}.$$

(16)

As shown in [31], $h_a\in {\mathcal{B}}_p$ and $L:=\underset{a\in \mathbb{D}}{sup}{\parallel h_a\parallel }_{{\mathcal{B}}_p}<\infty .$

A straightforward calculation shows that for $j\in \mathbb{N}$

$${\displaystyle \frac{d^j}{d{z}^j}}{h}_a\left(z\right)={\left(log{\displaystyle \frac{e}{1-{\left|a\right|}^2}}\right)}^{-1/p}{\displaystyle \frac{{\overline{a}}^j(j-1)!}{{(1-\overline{a}z)}^j}}.$$

In particular,

$$h_a^{\left(j\right)}\left(a\right)={\left(log{\displaystyle \frac{e}{1-{\left|a\right|}^2}}\right)}^{-1/p}{\displaystyle \frac{{\overline{a}}^j(j-1)!}{{(1-|a|}^2{)}^j}}.$$

Therefore

$$\begin{array}{c}L\parallel P_n^{\psi ,\phi }{\parallel }_{{\mathcal{B}}_p\to \mathcal{B}}\hfill \\ \ge \parallel P_n^{\psi ,\phi }{h}_{\phi \left(w\right)}{\parallel }_{\mathcal{B}}\hfill \\ \ge {(1-|w|}^2\left)\right|{\left(P_n^{\psi ,\phi }{h}_{\phi \left(w\right)}\right)}^{\prime }\left(w\right)|\hfill \\ ={(1-|w|}^2)|{\psi }_0^{\prime }\left(w\right)h_{\phi \left(w\right)}\left(\phi \left(w\right)\right)+\sum _{j=1}^{n+1}[{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)]h_{\phi \left(w\right)}^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ ={(1-|w|}^2)|{\psi }_0^{\prime }\left(w\right){(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}})}^{1-1/p}\hfill \\ +{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{-1/p}\sum _{j=1}^{n+1}{\displaystyle \frac{\phi {\left(w\right)}^j(j-1)![{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)]}{{(1-|\phi \left(w\right)|}^2{)}^j}}|.\hfill \end{array}$$

(17)

Hence

$$\begin{array}{cc}\hfill & {(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{1-1/p}\hfill \\ & \le L\parallel P_n^{\psi ,\phi }{\parallel }_{{\mathcal{B}}_p\to \mathcal{B}}+{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{-1/p}\sum _{j=1}^{n+1}{\displaystyle \frac{{(j-1)!(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^j}}\hfill \\ & \le L\parallel P_n^{\psi ,\phi }{\parallel }_{{\mathcal{B}}_p\to \mathcal{B}}+n!\sum _{j=1}^{n+1}{M}_j.\hfill \end{array}$$

(18)

Since $P_n^{\psi ,\phi }$ is bounded and ${\sum }_{j=1}^{n+1}{M}_j$ is finite, then by taking supremum of w over $\mathbb{D}$ we obtain

$$\begin{array}{c}\hfill \underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|{\left[log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right]}^{1-1/p}<\infty .\end{array}$$

(19)

□

As direct consequences of this characterization, we now obtain several corollaries covering important special cases.

Corollary 1.

Fix $n\in \mathbb{N}$, and let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi \in H\left(\mathbb{D}\right)$. Then the following are equivalent.

(a)

$D_{\psi ,\phi }^n:\mathcal{B}\to \mathcal{B}$ is bounded.

(b)

$D_{\psi ,\phi }^n:BMOA\to \mathcal{B}$ is bounded.

(c)

$D_{\psi ,\phi }^n:{\mathcal{B}}_p\to \mathcal{B}$ is bounded.

(d)

$\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^n}}=0,$ and $\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|\psi \left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^{n+1}}}=0.$

Corollary 2.

Let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi \in H\left(\mathbb{D}\right)$. Then the following are equivalent.

(a)

$W_{\psi ,\phi }:\mathcal{B}\to \mathcal{B}$ is bounded.

(b)

$W_{\psi ,\phi }:BMOA\to \mathcal{B}$ is bounded.

(c)

$\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}$ and $\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|\psi \left(w\right){\phi }^{\prime }\left(w\right)|}{1-{\left|\phi \left(w\right)\right|}^2}}<\infty .$

Corollary 3.

Let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi \in H\left(\mathbb{D}\right)$. Then the following are equivalent.

(a)

$W_{\psi ,\phi }:{\mathcal{B}}_p\to \mathcal{B}$ is bounded.

(b)

$\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left|\right)|{\psi }^{\prime }\left(w\right)|{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{1-1/p}$ and $\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|w|}^2)|\psi \left(w\right){\phi }^{\prime }\left(w\right)|}{1-{\left|\phi \left(w\right)\right|}^2}}<\infty .$

Corollary 4.

Fix $n\in \mathbb{N}$, and let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$. Then the following are equivalent.

(a)

$C_{\phi }{D}^n:\mathcal{B}\to \mathcal{B}$ is bounded.

(b)

$C_{\phi }{D}^n:BMOA\to \mathcal{B}$ is bounded.

(c)

$C_{\phi }{D}^n:{\mathcal{B}}_p\to \mathcal{B}$ is bounded.

(c)

$\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^{n+1}}}<\infty .$

Corollary 5.

Let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$. Then the following are equivalent.

(a)

$C_{\phi }:\mathcal{B}\to \mathcal{B}$ is bounded.

(b)

$C_{\phi }:BMOA\to \mathcal{B}$ is bounded.

(c)

$C_{\phi }:{\mathcal{B}}_p\to \mathcal{B}$ is bounded.

Proof. 

Note that, by Theorem 1, $C_{\phi }:\mathcal{B}(BMOA,\mathrm{or}{\mathcal{B}}_p)\to \mathcal{B}$ is bounded if and only if

$$\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\phi }^{\prime }\left(w\right)|}{1-{\left|\phi \left(w\right)\right|}^2}}<\infty .$$

Since $\phi \in \mathcal{S}\left(\mathbb{D}\right)$, the Schwarz–Pick Lemma ensures that this supremum is indeed finite. □

3. Compactness

We begin this section with a compactness criterion that will be used to characterize the compactness of sums of generalized weighted composition operators from Bloch, $BMOA$, and Besov spaces into the Bloch space. The proof follows standard arguments, similar to those in (Proposition 3.11, [3]).

Lemma 3.

Fix $n\in {\mathbb{N}}_0$, and let $X=\mathcal{B}$, $BMOA$, or ${\mathcal{B}}_p$. Suppose $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi =\{{\psi }_0,\dots ,{\psi }_n\}\subset \mathcal{H}\left(\mathbb{D}\right)$, such that $P_n^{\psi ,\phi }:X\to \mathcal{B}$ is bounded. Then, $P_n^{\psi ,\phi }$ is compact if and only if $\parallel P_n^{\psi ,\phi }{f}_k{\parallel }_X\to 0$ as $k\to \infty$ for any bounded sequence $\left\{f_k\right\}$ in X converging to 0 uniformly on compact subsets.

The following theorem addresses the compactness of sums of generalized weighted composition operators mapping Bloch spaces and $BMOA$ into the Bloch space.

Theorem 3.

Fix $n\in {\mathbb{N}}_0$, and let $\psi =\{{\psi }_0,\dots ,{\psi }_n\}\subset H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$ such that $P_n^{\psi ,\phi }$ from $\mathcal{B}$ (or $BMOA$) to $\mathcal{B}$) is bounded. Then the following statements are equivalent.

(a)

$P_n^{\psi ,\phi }:\mathcal{B}\to \mathcal{B}$ is compact.

(b)

$P_n^{\psi ,\phi }:\mathrm{BMOA}\to \mathcal{B}$ is compact.

(c)

$\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}=0$, $\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{{(1-|\phi \left(w\right)|}^2)}^j}}=0$, $1\le j\le n+1.$

Proof. 

That (a) implies (b) is an immediate consequence of the fact that $BMOA$ is continuously embedded into $\mathcal{B}$.

(b)⇒ (c) Assume that $P_n^{\psi ,\phi }:\mathrm{BMOA}\to \mathcal{B}$ compact. Let $\left\{w_m\right\}$ be a sequence in $\mathbb{D}$ such that $\left|\phi \right(w_m\left)\right|\to 1$. It follows that the sequence $f_{\phi \left(w_m\right)}$, introduced in the proof of Theorem 1, remains bounded in $BMOA$ and tends to 0 uniformly on compact subsets. By Lemma 3

$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}{(1-|w|}^2\left)\right|{\left(P_n^{\psi ,\phi }{f}_{\phi \left(m\right)}\right)}^{\prime }\left(w_m\right)|\le \underset{m\to \infty }{lim}{\parallel P_n^{\psi ,\phi }{f}_{\phi \left(m\right)}\parallel }_{\mathcal{B}}=0.\end{array}$$

(20)

Fix $k\in \{1,\dots ,n+1\}$ and recall that

$$f_{\phi \left(w_m\right)}\left(\phi \left(w_m\right)\right)=0\mathrm{and}\left|f_{\phi \left(w\right)}^{\left(t\right)}\left(\phi \left(w_m\right)\right)\right|={\displaystyle \frac{|\phi \left(w_m\right){|}^k}{(1-|\phi \left(w_m\right){{|}^2)}^k}}{\delta }_{t,k}$$

for $t\in \{1,\dots ,n+1\}$ Then, by equality (7), we have

$$\begin{array}{ccc}& & {\displaystyle \frac{(1-{\left|w_m\right|}^2\left)\right|{\psi }_j^{\prime }\left(w_m\right)+{\psi }_{j-1}\left(w_m\right){\phi }^{\prime }\left(w_m\right)|}{(1-|\phi \left(w_m\right){{|}^2)}^k}}=(1-{\left|w_m\right|}^2\left)\right|{\left(P_n^{\psi ,\phi }{f}_{\phi \left(w_m\right)}\right)}^{\prime }\left(w_m\right)|.\hfill \end{array}$$

(21)

Therefore, by letting $\left|\phi \right(w_m\left)\right|\to 1$ and using (20), we deduce that for each $k\in \{1,\dots ,n+1\}$

$$\underset{\left|\phi \right(w_m\left)\right|\to 1}{lim}{\displaystyle \frac{(1-{\left|w_m\right|}^2\left)\right|{\psi }_j^{\prime }\left(w_m\right)+{\psi }_{j-1}\left(w_m\right){\phi }^{\prime }\left(w_m\right)|}{(1-|\phi \left(w_m\right){{|}^2)}^k}}|=0.$$

(22)

Next we show that ${\displaystyle \underset{\left|\phi \right(w_m\left)\right|\to 1}{lim}(1-{\left|w_m\right|}^2\left)\right|{\psi }_0^{\prime }\left(w_m\right)|log\frac{e}{1-|\phi \left(w_m\right){|}^2}=0}$.

Consider a sequence $w_m\subset \mathbb{D}$ such that $\left|\phi \right(w_m\left)\right|\to 1$. In this case, the sequence $g_{\phi \left(w_m\right)}$ from the proof of Theorem 1 remains bounded in $BMOA$ and tends to 0 uniformly on compact subsets. Again, by Lemma 3,

$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}(1-{\left|w_m\right|}^2\left)\right|{\left(P_n^{\psi ,\phi }{g}_{\phi \left(w_m\right)}\right)}^{\prime }\left(w_m\right)|\le \underset{m\to \infty }{lim}{\parallel P_n^{\psi ,\phi }{g}_{\phi \left(w_m\right)}\parallel }_{\mathcal{B}}=0.\end{array}$$

(23)

Recalling that $g_{\phi \left(w_m\right)}\left(\phi \left(w_m\right)\right)=log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}$ and for $j=1\dots ,n+1$,

$$g_{\phi \left(w_m\right)}^{\left(j\right)}\left(\phi \left(w_m\right)\right)=\left[1+{\left(log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}\right)}^{-1}\sum _{t=1}^{j-1}{\displaystyle \frac{1}{t}}\right]{\displaystyle \frac{2{\overline{\phi \left(w_m\right)}}^j(j-1)!}{(1-|\phi \left(w_m\right){{|}^2)}^j}}.$$

Therefore, by (12), we obtain

$$\begin{array}{cc}\hfill & (1-{\left|w_m\right|}^2\left)\right|{\psi }_0^{\prime }\left(w_m\right)|log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}\hfill \\ & \le (1-{\left|w_m\right|}^2\left)\right|{\left(P_n^{\psi ,\phi }{f}_{\phi \left(m\right)}\right)}^{\prime }\left(w_m\right)|\hfill \\ & +\sum _{j=1}^{n+1}[\left(1+{\left(log{\displaystyle \frac{4e}{3}}\right)}^{-1}\sum _{t=1}^{j-1}{\displaystyle \frac{1}{t}}\right){\displaystyle \frac{2(j-1)!(1-{\left|w_m\right|}^2\left)\right|{\psi }_j^{\prime }\left(w_m\right)+{\psi }_{j-1}\left(w_m\right){\phi }^{\prime }\left(w_m\right)|}{(1-|\phi \left(w_m\right){{|}^2)}^j}}.\hfill \end{array}$$

Using (22) and (25) and letting $\left|\phi \right(w_m\left)\right|$ approach 1, we deduce that

$$\underset{\left|\phi \right(w_m\left)\right|\to 1}{lim}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w_m\right)|log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}=0.$$

(c)⇒ (a) Let $\left\{f_m\right\}$ be a bounded sequence in $\mathcal{B}$ converging uniformly to 0 on compact subsets of $\mathbb{D}$. Set $K=\underset{m}{sup}{\parallel f_m\parallel }_{\mathcal{B}}$. Then for $w\in \mathbb{D}$, we have

$$|f_m\left(\phi \left(w\right)\right)|\le log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}{\parallel f_m\parallel }_{\mathcal{B}}.$$

Moreover, by Theorem 5.5 in [36], there is a constant $C>0$ such that

$${{(1-|\phi \left(w\right)|}^2)}^j|f_m^{\left(j\right)}\left(\phi \left(w\right)\right)|\le C\parallel f_m{\parallel }_{\mathcal{B}}.$$

Fix $\epsilon >0$ and choose $\delta >0$ such that if $\delta <\left|\phi \right(w\left)\right|<1$, then

$${\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{{(1-|\phi \left(w\right)|}^2)}^j}}<{\displaystyle \frac{\epsilon }{(n+1)KC}},\mathrm{for}\mathrm{all}j=1,\dots ,n+1,$$

and

$${(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}<{\displaystyle \frac{\epsilon }{K}}.$$

Therefore

$$\begin{array}{cc}\hfill \parallel P_n^{\psi ,\phi }{f}_m{\parallel }_{\mathcal{B}}& \le |\left(P_n^{\psi ,\phi }{f}_m\right)\left(0\right)|+\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)\left|\right|f_m\left(\phi \left(w\right)\right)|\hfill \\ & +\sum _{j=1}^{n+1}\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)\left|\right|f_m^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ \hfill & \le |\left(P_n^{\psi ,\phi }{f}_m\right)\left(0\right)|+\underset{\left|\phi \right(w\left)\right|\le \delta }{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)\left|\right|f_m\left(\phi \left(w\right)\right)|\hfill \\ & +\parallel f_m{\parallel }_{\mathcal{B}}\underset{\left|\phi \right(w\left)\right|>\delta }{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\hfill \\ & +\sum _{j=1}^{n+1}\underset{\left|\phi \right(w\left)\right|\le \delta }{sup}{(1-|w|}^2\left)\right|{\psi }_j^{\prime }w+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)\left|\right|f^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ & +C\parallel f_m{\parallel }_{\mathcal{B}}\sum _{k=1}^{n+1}\underset{\left|\phi \right(w\left)\right|>\delta }{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{{(1-|\phi \left(w\right)|}^2)}^k}}\hfill \\ \hfill & <|\left(P_n^{\psi ,\phi }{f}_m\right)\left(0\right)|+\epsilon +\sum _{k=0}^{n+1}{M}_k\underset{\left|\phi \right(w\left)\right|\le \delta }{sup}\left|f_m^{\left(k\right)}\left(\phi \left(w\right)\right)\right|+\epsilon .\hfill \end{array}$$

Since $f_m\to 0$ uniformly on compact subsets of $\mathbb{D}$, Montel’s theorem implies that $f_m^{\left(k\right)}\to 0$ uniformly as well for all $k=1,\dots ,n+1$. As $\epsilon$ was arbitrary, we obtain

$$\underset{m\to \infty }{lim}{\parallel P_n^{\psi ,\phi }{f}_m\parallel }_{\mathcal{B}}=0.$$

□

Now, we turn our attention to the compactness of sums of generalized weighted composition operators from Besov Spaces to Bloch Spaces.

Theorem 4.

Fix $n\in {\mathbb{N}}_0$, and let $\psi =\{{\psi }_0,\dots ,{\psi }_n\}\subset H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$ such that $P_n^{\psi ,\phi }:{\mathcal{B}}_p\to \mathcal{B}$ is bounded. Then the following statements are equivalent.

(a)

$P_n^{\psi ,\phi }:{\mathcal{B}}_p\to \mathcal{B}$ is compact.

(b)

$\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{1-1/p}=0$, and

$\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{{(1-|\phi \left(w\right)|}^2)}^j}}=0$, $1\le j\le n+1$.

Proof. 

(b)⇒ (a) Let $f_m$ be a bounded sequence in ${\mathcal{B}}_p$ that converges uniformly to 0 on compact subsets of $\mathbb{D}$. Set $F:=\underset{m\in \mathbb{N}}{sup}{\parallel f_m\parallel }_{{\mathcal{B}}_p}$. Then for $w\in \mathbb{D}$, by Theorem 9 in [37], we have

$$|f_m\left(\phi \left(w\right)\right)|\le C{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{1-1/p}{\parallel f_m\parallel }_{{\mathcal{B}}_p}.$$

Furthermore, using the continuous embedding ${\mathcal{B}}_p\hookrightarrow \mathcal{B}$, it follows from Theorem 5.5 of [36] and (3) that, for every $k\in \mathbb{N}$, there exists a constant $C>0$ such that

$${{(1-|\phi \left(w\right)|}^2)}^j|f_m^{\left(j\right)}\left(\phi \left(w\right)\right)|\le C\parallel f_m{\parallel }_{{\mathcal{B}}_p}.$$

Fix $\epsilon >0$ and choose $\delta >0$ such that if $\delta <\left|\phi \right(w\left)\right|<1$, then

$${\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{{(1-|\phi \left(w\right)|}^2)}^j}}<{\displaystyle \frac{\epsilon }{(n+1)NC}},\mathrm{for}\mathrm{all}j=1,\dots ,n+1,$$

and

$${(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{1-1/p}<{\displaystyle \frac{\epsilon }{F}}.$$

It follows that

$$\begin{array}{cc}\hfill \parallel P_n^{\psi ,\phi }{f}_m{\parallel }_{\mathcal{B}}& \le |\left(P_n^{\psi ,\phi }{f}_m\right)\left(0\right)|+\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)\left|\right|f_m\left(\phi \left(w\right)\right)|\hfill \\ & +\sum _{j=1}^{n+1}\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)\left|\right|f_m^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ \hfill & \le |\left(P_n^{\psi ,\phi }{f}_m\right)\left(0\right)|+\underset{\left|\phi \right(w\left)\right|\le \delta }{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)\left|\right|f_m\left(\phi \left(w\right)\right)|\hfill \\ & +\parallel f_m{\parallel }_{{\mathcal{B}}_p}\underset{\left|\phi \right(w\left)\right|>\delta }{sup}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w\right)|{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{1-1/p}\hfill \\ & +\sum _{j=1}^{n+1}\underset{\left|\phi \right(w\left)\right|\le \delta }{sup}{(1-|w|}^2\left)\right|{\psi }_j^{\prime }w+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)\left|\right|f^{\left(j\right)}\left(\phi \left(w\right)\right)|\hfill \\ & +C\parallel f_m{\parallel }_{{\mathcal{B}}_p}\sum _{k=1}^{n+1}\underset{\left|\phi \right(w\left)\right|>\delta }{sup}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w\right)+{\psi }_{j-1}\left(w\right){\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^k}}\hfill \\ \hfill & <|\left(P_n^{\psi ,\phi }{f}_m\right)\left(0\right)|+\epsilon +\sum _{k=0}^{n+1}{M}_k\underset{\left|\phi \right(w\left)\right|\le \delta }{sup}\left|f_m^{\left(k\right)}\left(\phi \left(w\right)\right)\right|+\epsilon .\hfill \end{array}$$

Since $f_m$ converges to 0 uniformly on compacts subsets of $\mathbb{D}$, by Montel’s theorem, so does $f_m^{\left(k\right)}$ for all $k=1,\dots ,n+1$. Since $\epsilon$ is arbitrary, we obtain

$$\underset{m\to \infty }{lim}{\parallel P_n^{\psi ,\phi }{f}_m\parallel }_{\mathcal{B}}=0.$$

(a)⇒ (b) Assume that $P_n^{\psi ,\phi }:{\mathcal{B}}_p\to \mathcal{B}$ compact. Let $\left\{w_m\right\}$ be a sequence in $\mathbb{D}$ such that $\left|\phi \right(w_m\left)\right|\to 1$ and fix $k\in \{1,\dots ,n+1\}$. In Theorem 2, the sequence $f_{\phi \left(w_m\right)}$ is bounded in ${\mathcal{B}}_p$ and converges to 0 uniformly on compact subsets. By using a similar argument in the proof of the Theorem 3, we get

$$\underset{\left|\phi \right(w_m\left)\right|\to 1}{lim}{\displaystyle \frac{(1-{\left|w_m\right|}^2\left)\right|{\psi }_j^{\prime }\left(w_m\right)+{\psi }_{j-1}\left(w_m\right){\phi }^{\prime }\left(w_m\right)|}{(1-|\phi \left(w_m\right){{|}^2)}^k}}|=0.$$

(24)

Next we show that ${\displaystyle \underset{\left|\phi \right(w_m\left)\right|\to 1}{lim}(1-{\left|w_m\right|}^2\left)\right|{\psi }_0^{\prime }\left(w_m\right)|{\left(log\frac{e}{1-|\phi \left(w_m\right){|}^2}\right)}^{1-1/p}=0}$.

Let $\left\{w_m\right\}$ be a sequence in $\mathbb{D}$ such that $\left|\phi \right(w_m\left)\right|\to 1$. Then, the sequence $h_{\phi \left(w_m\right)}$ in the proof of Theorem 1 is bounded in ${\mathcal{B}}_p$ and converges to 0 uniformly on compact subsets. Again, by Lemma 3,

$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}(1-{\left|w_m\right|}^2\left)\right|{\left(P_n^{\psi ,\phi }{h}_{\phi \left(w_m\right)}\right)}^{\prime }\left(w_m\right)|\le \underset{m\to \infty }{lim}{\parallel P_n^{\psi ,\phi }{g}_{\phi \left(w_m\right)}\parallel }_{\mathcal{B}}=0.\end{array}$$

(25)

Recalling that $h_{\phi \left(w_m\right)}\left(\phi \left(w_m\right)\right)={\left[log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}\right]}^{1-1/p}$ and for $j=1,\dots ,n+1$,

$$h_{\phi \left(w_m\right)}^{\left(j\right)}\left(\phi \left(w_m\right)\right)={\left(log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}\right)}^{-1/p}{\displaystyle \frac{{\overline{\phi \left(w_m\right)}}^j(j-1)!}{(1-|\phi \left(w_m\right){{|}^2)}^j}}.$$

Therefore, by (18), we obtain

$$\begin{array}{cc}\hfill & (1-{\left|w_m\right|}^2\left)\right|{\psi }_0^{\prime }\left(w_m\right)|{(log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}})}^{1-1/p}\hfill \\ & \le (1-{\left|w_m\right|}^2\left)\right|{\left(P_n^{\psi ,\phi }{f}_{\phi \left(m\right)}\right)}^{\prime }\left(w_m\right)|\hfill \\ & +{\left(log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}\right)}^{-1/p}\sum _{j=1}^{n+1}{\displaystyle \frac{{(j-1)!(1-|w|}^2\left)\right|{\psi }_j^{\prime }\left(w_m\right)+{\psi }_{j-1}\left(w_m\right){\phi }^{\prime }\left(w_m\right)|}{(1-|\phi \left(w_m\right){{|}^2)}^j}}.\hfill \end{array}$$

Using (24), (25) and letting $\left|\phi \right(w_m\left)\right|$ approach 1, we deduce that

$$\underset{\left|\phi \right(w_m\left)\right|\to 1}{lim}{(1-|w|}^2\left)\right|{\psi }_0^{\prime }\left(w_m\right)|{(log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}})}^{1-1/p}=0.$$

□

As an immediate application of this compactness criterion, we derive several corollaries that describe the compactness of important special cases.

Corollary 6.

Fix $n\in \mathbb{N}$, and let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi \in H\left(\mathbb{D}\right)$. $D_{\psi ,\phi }^n:\mathcal{B}({\mathcal{B}}_{p}or$ BMOA $)\to \mathcal{B}$ bounded. Then the following are equivalent.

(a)

$D_{\psi ,\phi }^n:\mathcal{B}\to \mathcal{B}$ is compact.

(b)

$D_{\psi ,\phi }^n:BMOA\to \mathcal{B}$ is compact.

(c)

$D_{\psi ,\phi }^n:{\mathcal{B}}_p\to \mathcal{B}$ is compact.

(d)

$\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\psi }^{\prime }\left(w\right)|}{{{(1-|\phi \left(w\right)|}^2)}^n}}=0,$ and $\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{\displaystyle \frac{{(1-|w|}^2)|\psi \left(w\right){\phi }^{\prime }\left(w\right)|}{{{(1-|\phi \left(w\right)|}^2)}^{n+1}}}=0.$

Corollary 7.

Let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi \in H\left(\mathbb{D}\right)$. Suppose that $W_{\psi ,\phi }:\mathcal{B}\left(BMOA\right)\to \mathcal{B}$ is bounded. Then the following are equivalent.

(a)

$W_{\psi ,\phi }:\mathcal{B}\to \mathcal{B}$ is compact.

(b)

$W_{\psi ,\phi }:BMOA\to \mathcal{B}$ is compact.

(c)

$\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{(1-|w|}^2\left|\right)|{\psi }^{\prime }\left(w\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}=0$ and $\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{\displaystyle \frac{{(1-|w|}^2)|\psi \left(w\right){\phi }^{\prime }\left(w\right)|}{1-{\left|\phi \left(w\right)\right|}^2}}=0.$

Corollary 8.

Let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$ and $\psi \in H\left(\mathbb{D}\right)$. Suppose that $W_{\psi ,\phi }:{\mathcal{B}}_p\to \mathcal{B}$ is bounded. Then the following are equivalent.

(a)

$W_{\psi ,\phi }:{\mathcal{B}}_p\to \mathcal{B}$ is compact.

(b)

$\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{(1-|w|}^2\left|\right)|{\psi }^{\prime }\left(w\right)|{\left(log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\right)}^{1-1/p}=0$ and $\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{\displaystyle \frac{{(1-|w|}^2)|\psi \left(w\right){\phi }^{\prime }\left(w\right)|}{1-{\left|\phi \left(w\right)\right|}^2}}=0.$

Corollary 9.

Fix $n\in {\mathbb{N}}_0$, and let $\phi \in \mathcal{S}\left(\mathbb{D}\right)$. Suppose that $C_{\phi }{D}^n:BMOA\left(\mathcal{B}or{\mathcal{B}}_p\right)\to \mathcal{B}$ is bounded.

Then the following are equivalent.

(a)

$C_{\phi }{D}^n:\mathcal{B}\to \mathcal{B}$ is compact.

(b)

$C_{\phi }{D}^n:BMOA\to \mathcal{B}$ is compact.

(c)

$C_{\phi }{D}^n:{\mathcal{B}}_p\to \mathcal{B}$ is compact.

(d)

$\underset{\left|\phi \right(w\left)\right|\to 1}{lim}{\displaystyle \frac{{(1-|w|}^2\left)\right|{\phi }^{\prime }\left(w\right)|}{{(1-|\phi \left(w\right)|}^2{)}^{n+1}}}=0.$

4. Conclusions

In summary, the interplay between analytic function spaces and operator theory continues to be a central theme in complex analysis. By studying generalized operators such as $P_n^{\psi ,\phi }$, which unify composition, differentiation, and multiplication effects, we gain deeper insights into the boundedness and compactness criteria across various spaces. The aim of this paper is to establish unified conditions under which these operators map from $BMOA$ and analytic Besov spaces into the Bloch space. Our results generalize and extend several known theorems, offering a broader framework for understanding analytic operator behavior in function space theory.