Natural metrics and composition operators in generalized hyperbolic function spaces

Abstract

In this paper, we define some generalized hyperbolic function classes. We also introduce natural metrics in the generalized hyperbolic $(p,\alpha )$-Bloch and in the generalized hyperbolic $Q^{\ast }(p,s)$ classes. These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, boundedness and compactness the composition operators $C_{\varphi }$ acting from the generalized hyperbolic $(p,\alpha )$-Bloch class to the class $Q^{\ast }(p,s)$ are characterized by conditions depending on an analytic self-map $\varphi :\mathbb{D}\to \mathbb{D}$.

MSC:47B38, 46E15.

1 Introduction

Let $\mathbb{D}=\{z:|z|<1\}$ be the open unit disc of the complex plane $\mathbb{C}$, $\partial \mathbb{D}$ its boundary. Let $\mathcal{H}(\mathbb{D})$ denote the space of all analytic functions in $\mathbb{D}$ and let $B(\mathbb{D})$ be the subset of $\mathcal{H}(\mathbb{D})$ consisting of those $f\in \mathcal{H}(\mathbb{D})$ for which $|f(z)|<1$ for all $z\in \mathbb{D}$. Also, $dA(z)$ be the normalized area measure on $\mathbb{D}$ so that $A(\mathbb{D})\equiv 1$. The usual α-Bloch spaces ${\mathcal{B}}_{\alpha }$ and ${\mathcal{B}}_{\alpha ,0}$ are defined as the sets of those $f\in \mathcal{H}(\mathbb{D})$ for which

and

respectively. Now, we will give the following definition:

Definition 1.1 The $(p,\alpha )$-Bloch spaces ${\mathcal{B}}_{p,\alpha }$ and ${\mathcal{B}}_{p,\alpha ,0}$ are defined as the sets of those $f\in \mathcal{H}(\mathbb{D})$ for which

and

where $0<p,\alpha <\mathrm{\infty }$.

Remark 1.1 The definition of $(p,\alpha )$-Bloch spaces is introduced in the present paper for the first time. One should note that, if we put $p=2$ in Definition 1.1, we will obtain the spaces ${\mathcal{B}}_{\alpha }$ and ${\mathcal{B}}_{\alpha ,0}$.

Remark 1.2$(p,\alpha )$-Bloch space is very useful in some calculations in this paper and it can be also used to study some other operators like integral operators (see [12]).

If $(X,d)$ is a metric space, we denote the open and closed balls with center x and radius $r>0$ by $B(x,r):=\{y\in X:d(y,x)<r\}$ and $\overline{B}(x,r):=\{y\in X:d(y,x)=r\}$, respectively. The well-known hyperbolic derivative is defined by $f^{\ast }(z)=\frac{|f^{\prime }(z)|}{1-{|f(z)|}^2}$ of $f\in B(\mathbb{D})$ and the hyperbolic distance is given by $\rho (f(z),0):=\frac{1}{2}log(\frac{1+|f(z)|}{1-|f(z)|})$ between $f(z)$ and zero.

A function $f\in B(\mathbb{D})$ is said to belong to the hyperbolic α-Bloch class ${\mathcal{B}}_{\alpha }^{\ast }$ if

$${\parallel f\parallel }_{{\mathcal{B}}_{\alpha }^{\ast }}=\underset{z\in \mathbb{D}}{sup}{f}^{\ast }(z){(1-{|z|}^2)}^{\alpha }<\mathrm{\infty }.$$

The little hyperbolic Bloch-type class ${\mathcal{B}}_{\alpha ,0}^{\ast }$ consists of all $f\in {\mathcal{B}}_{\alpha }^{\ast }$ such that

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

The Schwarz-Pick lemma implies ${\mathcal{B}}_{\alpha }^{\ast }=B(\mathbb{D})$ for all $\alpha \ge 1$ with ${\parallel f\parallel }_{{\mathcal{B}}_{\alpha }^{\ast }}\le 1$, and therefore, the hyperbolic α-Bloch classes are of interest only when $0<\alpha <1$.

It is obvious that ${\mathcal{B}}_{\alpha }^{\ast }$ is not a linear space since the sum of two functions in $B(\mathbb{D})$ does not necessarily belong to $B(\mathbb{D})$.

Now, let $0<p<\mathrm{\infty }$, we define the hyperbolic derivative by $f_p^{\ast }(z)=\frac{p}{2}\frac{{|f(z)|}^{\frac{p}{2}-1}|f^{\prime }(z)|}{1-{|f(z)|}^p}$ of $f\in B(\mathbb{D})$. When $p=2$, we obtain the usual hyperbolic derivative as defined above.

A function $f\in B(\mathbb{D})$ is said to belong to the generalized hyperbolic $(p,\alpha )$-Bloch class ${\mathcal{B}}_{p,\alpha }^{\ast }$ if

$${\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}=\underset{z\in \mathbb{D}}{sup}{f}_p^{\ast }(z){(1-{|z|}^2)}^{\alpha }<\mathrm{\infty }.$$

The little generalized $(p,\alpha )$-hyperbolic Bloch-type class ${\mathcal{B}}_{p,\alpha ,0}^{\ast }$ consists of all $f\in {\mathcal{B}}_{p,\alpha }^{\ast }$ such that

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

Let the Green’s function of $\mathbb{D}$ be defined as $g(z,a)=log\frac{1}{|{\phi }_a(z)|}$, where ${\phi }_a(z)=\frac{a-z}{1-\overline{a}z}$ is the Möbius transformation related to the point $a\in \mathbb{D}$. For $0<p,s<\mathrm{\infty }$, the hyperbolic class $Q^{\ast }(p,s)$ consists of those functions $f\in B(\mathbb{D})$ for which

$${\parallel f\parallel }_{Q^{\ast }(p,s)}^p=\underset{a\in \mathbb{D}}{sup}{\int }_{\mathbb{D}}{(f_p^{\ast }(z))}^2{g}^s(z,a)dA(z)<\mathrm{\infty }.$$

Moreover, we say that $f\in Q^{\ast }(p,s)$ belongs to the class $Q^{\ast }(p,s,0)$ if

$$\underset{|a|\to 1}{lim}{\int }_{\mathbb{D}}{(f_p^{\ast }(z))}^2{g}^s(z,a)dA(z)=0.$$

When $p=2$, we obtain the usual hyperbolic Q class as studied in [10, 11, 14].

Remark 1.3 The Schwarz-Pick lemma implies that ${\mathcal{B}}_{p,\alpha }^{\ast }=B(\mathbb{D})$ for all $\alpha \ge 1$ with ${\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}\le 1$ and therefore, the generalized hyperbolic $(p,\alpha )$-classes are of interest only when $0<\alpha <1$. Also $Q^{\ast }(p,s)=B(\mathbb{D})$ for all $s>1$, and hence, the generalized hyperbolic $Q(p,s)$-classes will be considered when $0\le s\le 1$.

For any holomorphic self-mapping ϕ of $\mathbb{D}$, the symbol ϕ induces a linear composition operator $C_{\varphi }(f)=f\circ \varphi$ from $\mathcal{H}(\mathbb{D})$ or $B(\mathbb{D})$ into itself. The study of a composition operator $C_{\varphi }$ acting on the spaces of analytic functions has engaged many analysts for many years (see, e.g., [1–8, 11, 13, 16] and others).

Yamashita was probably the first to consider systematically hyperbolic function classes. He introduced and studied hyperbolic Hardy, BMOA and Dirichlet classes in [18–20] and others. More recently, Smith studied inner functions in the hyperbolic little Bloch-class [15], and the hyperbolic counterparts of the $Q_p$ spaces were studied by Li in [10] and Li et al. in [11]. Further, hyperbolic $Q_p$ classes and composition operators were studied by Pérez-González et al. in [14].

In this paper, we will study the generalized hyperbolic $(p,\alpha )$-Bloch classes ${\mathcal{B}}_{p,\alpha }^{\ast }$ and the hyperbolic $Q^{\ast }(p,s)$ type classes. We will also give some results to characterize Lipschitz continuous and compact composition operators mapping from the generalized hyperbolic $(p,\alpha )$-Bloch class ${\mathcal{B}}_{p,\alpha }^{\ast }$ to $Q^{\ast }(p,s)$ classes by conditions depending on the symbol ϕ only. Thus, the results are generalizations of the recent results of Pérez-González, Rättyä and Taskinen [14].

Recall that a linear operator $T:X\to Y$ is said to be bounded if there exists a constant $C>0$ such that ${\parallel T(f)\parallel }_Y\le C{\parallel f\parallel }_X$ for all maps $f\in X$. By elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the Lipschitz-continuity. Moreover, $T:X\to Y$ is said to be compact if it takes bounded sets in X to sets in Y which have compact closure. For Banach spaces X and Y contained in $B(\mathbb{D})$ or $\mathcal{H}(\mathbb{D})$, $T:X\to Y$ is compact if and only if for each bounded sequence $(x_n)\in X$, the sequence $(T{x}_n)\in Y$ contains a subsequence converging to a function $f\in Y$.

Throughout this paper, C stands for absolute constants which may indicate different constants from one occurrence to the next.

The following lemma follows by standard arguments similar to those outlined in [17]. Hence we omit the proof.

Lemma 1.1 Assume ϕ is a holomorphic mapping from$\mathbb{D}$into itself. Let$0<p,s<\mathrm{\infty }$, and$0<\alpha <\mathrm{\infty }$. Then$C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$is compact if and only if for any bounded sequence${(f_n)}_{n\in \mathbb{N}}\in {\mathcal{B}}_{p,\alpha }^{\ast }$which converges to zero uniformly on compact subsets of$\mathbb{D}$as$n\to \mathrm{\infty }$, we have${lim}_{n\to \mathrm{\infty }}{\parallel C_{\varphi }{f}_n\parallel }_{Q^{\ast }(p,s)}=0$.

Using the standard arguments similar to those outlined in Lemma 1 of [9], we have the following lemma:

Lemma 1.2 Let$0<\alpha <\mathrm{\infty }$, then there exist two functions$f,g\in {\mathcal{B}}_{p,\alpha }^{\ast }$such that for some constant C,

$$(|f_p^{\ast }(z)|+|g_p^{\ast }(z)|){(1-{|z|}^2)}^{\alpha }\ge C>0,\mathit{\text{for each }}z\in \mathbb{D}.$$

2 Natural metrics in ${\mathcal{B}}_{p,\alpha }^{\ast }$ and $Q^{\ast }(p,s)$ classes

In this section we introduce natural metrics on generalized hyperbolic α-Bloch classes ${\mathcal{B}}_{p,\alpha }^{\ast }$ and the classes $Q^{\ast }(p,s)$.

Let $0<p,s<\mathrm{\infty }$, and $0<\alpha <1$. First, we can find a natural metric in ${\mathcal{B}}_{p,\alpha }^{\ast }$ (see [14]) by defining

$$d(f,g;{\mathcal{B}}_{p,\alpha }^{\ast }):=d_{{\mathcal{B}}_{p,\alpha }^{\ast }}(f,g)+{\parallel f-g\parallel }_{{\mathcal{B}}_{p,\alpha }}+{|f(0)-g(0)|}^{\frac{p}{2}},$$

(1)

where

$$d_{{\mathcal{B}}_{p,\alpha }^{\ast }}(f,g):=\underset{z\in \mathbb{D}}{sup}|\frac{f^{\prime }(z){|f(z)|}^{\frac{p}{2}-1}}{1-{|f(z)|}^p}-\frac{g^{\prime }(z){|g(z)|}^{\frac{p}{2}-1}}{1-{|g(z)|}^p}|{(1-{|z|}^2)}^{\alpha }.$$

For $f,g\in Q^{\ast }(p,s)$, define their distance by

$$d(f,g;Q^{\ast }(p,s)):=d_{Q^{\ast }}(f,g)+{\parallel f-g\parallel }_{Q(p,s)}+{|f(0)-g(0)|}^{\frac{p}{2}},$$

where

$$d_{Q^{\ast }}(f,g):={(\frac{p}{2}\underset{z\in \mathbb{D}}{sup}{\int }_{\mathbb{D}}{|\frac{f^{\prime }(z){|f(z)|}^{\frac{p}{2}-1}}{1-{|f(z)|}^p}-\frac{g^{\prime }(z){|g(z)|}^{\frac{p}{2}-1}}{1-{|g(z)|}^p}|}^2{g}^s(z,a)dA(z))}^{\frac{1}{2}}.$$

Now, we give a characterization of the complete metric space $d(\cdot ,\cdot ;{\mathcal{B}}_{p,\alpha }^{\ast })$.

Proposition 2.1 The class${\mathcal{B}}_{p,\alpha }^{\ast }$equipped with the metric$d(\cdot ,\cdot ;{\mathcal{B}}_{p,\alpha }^{\ast })$is a complete metric space. Moreover, ${\mathcal{B}}_{p,\alpha ,0}^{\ast }$is a closed (and therefore complete) subspace of${\mathcal{B}}_{p,\alpha }^{\ast }$.

Proof Clearly $d(f,g;{\mathcal{B}}_{p,\alpha }^{\ast })\ge 0$, $d(f,g,{\mathcal{B}}_{p,\alpha }^{\ast })=d(g,f;{\mathcal{B}}_{p,\alpha }^{\ast })$. Also,

$$d(f,h;{\mathcal{B}}_{p,\alpha }^{\ast })\le d(f,g;{\mathcal{B}}_{p,\alpha }^{\ast })+d(g,h;{\mathcal{B}}_{p,\alpha }^{\ast }).$$

Moreover, $d(f,f;{\mathcal{B}}_{p,\alpha }^{\ast })=0$ for all $f,g,h\in {\mathcal{B}}_{p,\alpha }^{\ast }$.

It follows from the presence of the usual $(p,\alpha )$-Bloch term that $d(f,g;{\mathcal{B}}_{p,\alpha }^{\ast })=0$ implies $f=g$. Hence, $({\mathcal{B}}_{p,\alpha }^{\ast },d)$ is a metric space. Let ${(f_n)}_{n=1}^{\mathrm{\infty }}$ be a Cauchy sequence in the metric space $({\mathcal{B}}_{p,\alpha }^{\ast },d)$, that is, for any $\epsilon >0$, there is an $N=N(\epsilon )\in \mathbb{N}$ such that

$$d(f_n,f_m;{\mathcal{B}}_{p,\alpha }^{\ast })<\epsilon$$

for all $n,m>N$. Since $(f_n)\subset B(\mathbb{D})$, the family $(f_n)$ is uniformly bounded and hence normal in $\mathbb{D}$. Therefore, there exist $f\in B(\mathbb{D})$ and a subsequence ${(f_{n_j})}_{j=1}^{\mathrm{\infty }}$ such that $f_{n_j}$ converges to f uniformly on compact subsets, and by the Cauchy formula, the same also holds for the derivatives. Let $m>N$. Then the uniform convergence yields

(2)

for all $z\in \mathbb{D}$, and it follows that

$${\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}\le {\parallel f_m\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}+\epsilon .$$

Thus, $f\in {\mathcal{B}}_{p,\alpha }^{\ast }$ as desired. Moreover, (2) and the completeness of the usual $(p,\alpha )$-Bloch imply that ${(f_n)}_{n=1}^{\mathrm{\infty }}$ converges to f with respect to the metric d. The second part of the assertion follows by (2). □

Next, we give a characterization of the complete metric space $d(\cdot ,\cdot ;Q^{\ast }(p,s))$.

Proposition 2.2 The class$Q^{\ast }(p,s)$equipped with the metric$d(\cdot ,\cdot ;Q^{\ast }(p,s))$is a complete metric space. Moreover, $Q^{\ast }(p,s,0)$is a closed (and therefore complete) subspace of$Q^{\ast }(p,s)$.

Proof For $f,g,h\in Q^{\ast }(p,s)$, then clearly

• $d(f,g;Q^{\ast }(p,s))\ge 0$,

• $d(f,f;Q^{\ast }(p,s))=0$,

• $d(f,g;Q^{\ast }(p,s))=0$ implies $f=g$,

• $d(f,g;Q^{\ast }(p,s))=d(g,f;Q^{\ast }(p,s))$,

• $d(f,h;Q^{\ast }(p,s))\le d(f,g;Q^{\ast }(p,s))+d(g,h;Q^{\ast }(p,s))$.

Hence, d is metric on $Q^{\ast }(p,s)$.

For the completeness proof, let ${(f_n)}_{n=1}^{\mathrm{\infty }}$ be a Cauchy sequence in the metric space $(Q^{\ast }(p,s),d)$, that is, for any $\epsilon >0$ there is an $N=N(\epsilon )\in \mathbb{N}$ such that $d(f_n,f_m;Q^{\ast }(p,s))<\epsilon$, for all $n,m>N$. Since $f_n\in B(\mathbb{D})$ such that $f_n$ converges to f uniformly on compact subsets of $\mathbb{D}$. Let $m>N$ and $0<r<1$. Then Fatou’s lemma yields

and by letting $r\to 1^{-}$, it follows that

$${\int }_{\mathbb{D}}{(f_p^{\ast }(z))}^2{g}^s(z,a)dA(z)\le 2{\epsilon }^2+2{\int }_{\mathbb{D}}{\left|\frac{{|f_m(z)|}^{\frac{p}{2}-1}{f}_m^{\prime }(z)}{1-{|f_m(z)|}^p}\right|}^2{g}^s(z,a)dA(z).$$

(3)

This yields

$${\parallel f\parallel }_{Q^{\ast }(p,s)}^p\le 2{\epsilon }^2+2{\parallel f_m\parallel }_{Q^{\ast }(p,s)}^2,$$

and thus $f\in Q^{\ast }(p,s)$. We also find that $f_n\to f$ with respect to the metric of $Q^{\ast }(p,s)$. The second part of the assertion follows by (3). □

3 Lipschitz continuous and compactness of $C_{\varphi }$

Theorem 3.1 Let$0<p<\mathrm{\infty }$, $0\le s\le 1$, and$0<\alpha \le 1$. Assume that ϕ is a holomorphic mapping from$\mathbb{D}$into itself. Then the following statements are equivalent:

• (i) $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$ is bounded;

• (ii) $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$ is Lipschitz continuous;

• (iii) ${sup}_{a\in \mathbb{D}}{\int }_{\mathbb{D}}\frac{{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{g}^s(z,a)dA(z)<\mathrm{\infty }$.

Proof First, assume that (i) holds, then there exists a constant C such that

$${\parallel C_{\varphi }f\parallel }_{Q^{\ast }(p,s)}\le C{\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }},\text{for all }f\in {\mathcal{B}}_{p,\alpha }^{\ast }.$$

For given $f\in {\mathcal{B}}_{p,\alpha }^{\ast }$, the function $f_t(z)=f(tz)$, where $0<t<1$, belongs to ${\mathcal{B}}_{p,\alpha }^{\ast }$ with the property ${\parallel f_t\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}\le {\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}$. Let f, g be the functions from Lemma 1.2 such that

$$\frac{1}{{(1-{|z|}^2)}^{\alpha }}\le |f_p^{\ast }(z)|+|g_p^{\ast }(z)|,$$

for all $z\in \mathbb{D}$, so that

$$\frac{|{\varphi }^{\prime }(z)|}{{(1-|\varphi (z)|)}^{\alpha }}\le {(f\circ \varphi )}^{\ast }(z)+{(g\circ \varphi )}^{\ast }(z).$$

Thus,

This estimate together with the Fatou’s lemma implies (iii).

Conversely, assuming that (iii) holds and that $f\in {\mathcal{B}}_{p,\alpha }^{\ast }$, we see that

Hence, it follows that (i) holds.

(ii) ⟺ (iii). Assume first that $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$ is Lipschitz continuous, that is, there exists a positive constant C such that

$$d(f\circ \varphi ,g\circ \varphi ;Q^{\ast }(p,s))\le Cd(f,g;{\mathcal{B}}_{p,\alpha }^{\ast }),\text{for all }f,g\in {\mathcal{B}}_{p,\alpha }^{\ast }.$$

Taking $g=0$, this implies

$${\parallel f\circ \varphi \parallel }_{Q^{\ast }(p,s)}\le C({\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}+{\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }}+{|f(0)|}^{\frac{p}{2}}),\text{for all }f\in {\mathcal{B}}_{p,\alpha }^{\ast }.$$

(4)

The assertion (iii) for $\alpha =1$ follows by choosing $f(z)=z$ in (4). If $0<\alpha <1$, then

$$\begin{array}{rcl}\frac{2}{p}{|f(z)|}^{\frac{p}{2}}& =& |{\int }_0^z{|f(t)|}^{\frac{p}{2}}{f}^{\prime }(t)dt+{(f(0))}^{\frac{p}{2}}|\\ \le & {\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }}{\int }_0^{|z|}\frac{dx}{{(1-x^2)}^{\alpha }}+{|f(0)|}^{\frac{p}{2}}\\ \le & \frac{{\parallel f\parallel }_{{\mathcal{B}}_{\alpha }}}{(1-\alpha )}+{|f(0)|}^{\frac{p}{2}},\end{array}$$

this yields

$$\frac{2}{p}{|f(\varphi (0))-g(\varphi (0))|}^{\frac{p}{2}}\le \frac{{\parallel f-g\parallel }_{{\mathcal{B}}_{p,\alpha }}}{(1-\alpha )}+{|f(0)-g(0)|}^{\frac{p}{2}}.$$

Moreover, Lemma 1.2 implies the existence of $f,g\in {\mathcal{B}}_{p,\alpha }^{\ast }$ such that

$$|f_p^{\ast }(z)+g_p^{\ast }(z)|{(1-{|z|}^2)}^{\alpha }\ge C>0,\text{for all }z\in \mathbb{D}.$$

(5)

Combining (4) and (5), we obtain

for which the assertion (iii) follows.

Assume now that (iii) is satisfied, we have

$$\begin{array}{rcl}d(f\circ \varphi ,g\circ \varphi ;Q^{\ast }(p,s))& =& d_{Q_{(p,s)}^{\ast }}(f\circ \varphi ,g\circ \varphi )+{\parallel f\circ \varphi -g\circ \varphi \parallel }_{Q(p,s)}\\ +{|f(\varphi (0))-g(\varphi (0))|}^{\frac{p}{2}}\\ \le & d_{{\mathcal{B}}_{p,\alpha }^{\ast }}(f,g){(\underset{a\in \mathbb{D}}{sup}{\int }_{\mathbb{D}}\frac{{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{g}^s(z,a)dA(z))}^{\frac{1}{2}}\\ +{\parallel f-g\parallel }_{{\mathcal{B}}_{p,\alpha }}{(\underset{a\in \mathbb{D}}{sup}{\int }_{\mathbb{D}}\frac{{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{g}^s(z,a)dA(z))}^{\frac{1}{2}}\\ +\frac{{\parallel f-g\parallel }_{{\mathcal{B}}_{p,\alpha }}}{(1-\alpha )}+{|f(0)-g(0)|}^{\frac{p}{2}}\\ \le & Cd(f,g;{\mathcal{B}}_{p,\alpha }^{\ast }).\end{array}$$

Thus $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$ is Lipschitz continuous and the proof is completed. □

Remark 3.1 We know that a composition operator $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$ is said to be bounded if there is a positive constant C such that ${\parallel C_{\varphi }f\parallel }_{Q^{\ast }(p,s)}\le C{\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}$ for all $f\in {\mathcal{B}}_{p,\alpha }^{\ast }$. Theorem 3.1 shows that $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$ is bounded if and only if it is Lipschitz-continuous, that is, if there exists a positive constant C such that

$$d(f\circ \varphi ,g\circ \varphi ;Q^{\ast }(p,s))\le Cd(f,g;{\mathcal{B}}_{p,\alpha }^{\ast }),\text{for all }f,g\in {\mathcal{B}}_{p,\alpha }^{\ast }.$$

By elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the Lipschitz-continuity. So, our result for composition operators in hyperbolic spaces is the correct and natural generalization of the linear operator theory.

Recall that a composition operator $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$ is compact if it maps any ball in ${\mathcal{B}}_{p,\alpha }^{\ast }$ onto a precompact set in $Q^{\ast }(p,s)$.

The following observation is sometimes useful.

Proposition 3.1 Let$0<p<\mathrm{\infty }$, $0\le s\le 1$and$0<\alpha \le 1$. Assume that ϕ is a holomorphic mapping from$\mathbb{D}$into itself. If$C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$is compact, it maps closed balls onto compact sets.

Proof If $B\subset {\mathcal{B}}_{p,\alpha }^{\ast }$ is a closed ball and $g\in Q^{\ast }(p,s)$ belongs to the closure of $C_{\varphi }(B)$, we can find a sequence ${(f_n)}_{n=1}^{\mathrm{\infty }}\subset B$ such that $f_n\circ \varphi$ converges to $g\in Q^{\ast }(p,s)$ as $n\to \mathrm{\infty }$. But ${(f_n)}_{n=1}^{\mathrm{\infty }}$ is a normal family, hence it has a subsequence ${(f_{n_j})}_{j=1}^{\mathrm{\infty }}$ converging uniformly on the compact subsets of $\mathbb{D}$ to an analytic function f. As in earlier arguments of Proposition 2.1 in [14], we get a positive estimate which shows that f must belong to the closed ball B. On the other hand, also the sequence ${(f_{n_j}\circ \varphi )}_{j=1}^{\mathrm{\infty }}$ converges uniformly on compact subsets to an analytic function, which is $g\in Q^{\ast }(p,s)$. We get $g=f\circ \varphi$, i.e., g belongs to $C_{\varphi }(B)$. Thus, this set is closed and also compact. □

Compactness of composition operators can be characterized in full analogy with the linear case.

Theorem 3.2 Let$0<p<\mathrm{\infty }$, $0\le s\le 1$, and$0<\alpha \le 1$. Assume that ϕ is a holomorphic mapping from$\mathbb{D}$into itself. Then the following statements are equivalent:

• (i) $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to Q^{\ast }(p,s)$ is compact.(ii)

  $$\underset{r\to 1^{-}}{lim}\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi |\ge r_j}\frac{{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{g}^s(z,a)dA(z)=0.$$

Proof We first assume that (ii) holds. Let $B:=\overline{B}(g,\delta )\subset {\mathcal{B}}_{p,\alpha }^{\ast }$, where $g\in {\mathcal{B}}_{p,\alpha }^{\ast }$ and $\delta >0$, be a closed ball, and let ${(f_n)}_{n=1}^{\mathrm{\infty }}\subset B$ be any sequence. We show that its image has a convergent subsequence in $Q^{\ast }(p,s)$, which proves the compactness of $C_{\varphi }$ by definition.

Again, ${(f_n)}_{n=1}^{\mathrm{\infty }}\subset B(\mathbb{D})$ is a normal family, hence there is a subsequence ${(f_{n_j})}_{j=1}^{\mathrm{\infty }}$ which converges uniformly on the compact subsets of $\mathbb{D}$ to an analytic function f. By the Cauchy formula for the derivative of an analytic function, also the sequence ${(f_{n_j}^{\mathrm{\prime }})}_{j=1}^{\mathrm{\infty }}$ converges uniformly to $f^{\mathrm{\prime }}$. It follows that also the sequences ${(f_{n_j}\circ \varphi )}_{j=1}^{\mathrm{\infty }}$ and ${(f_{n_j}^{\mathrm{\prime }}\circ \varphi )}_{j=1}^{\mathrm{\infty }}$ converge uniformly on the compact subsets of $\mathbb{D}$ to $f\circ \varphi$ and $f^{\prime }\circ \varphi$, respectively. Moreover, $f\in B\subset {\mathcal{B}}_{p,\alpha }^{\ast }$ since for any fixed R, $0<R<1$, the uniform convergence yields

Hence, $d(f,g;{\mathcal{B}}_{p,\alpha }^{\ast })\le \delta$.

Let $\epsilon >0$. Since (ii) is satisfied, we may fix r, $0<r<1$, such that

$$\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\ge r}\frac{{|\varphi (z)|}^{p-2}{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{g}^s(z,a)dA(z)\le \epsilon .$$

By the uniform convergence, we may fix $N_1\in \mathbb{N}$ such that

$$|f_{n_j}\circ \varphi (0)-f\circ \varphi (0)|\le \epsilon ,\text{for all }j\ge N_1.$$

(6)

The condition (ii) is known to imply the compactness of $C_{\varphi }:{\mathcal{B}}_{p,\alpha }\to Q(p,s)$, hence possibly to passing once more to a subsequence and adjusting the notations, we may assume that

$${\parallel f_{n_j}\circ \varphi -f\circ \varphi \parallel }_{Q(p,s)}\le \epsilon ,\text{for all }j\ge N_2,\text{ for some }{N}_2\in \mathbb{N}.$$

(7)

Now let

$$I_1(a,r)=\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\ge r}{[{(f_{n_j}\circ \varphi )}_p^{\ast }(z)-{(f\circ \varphi )}_p^{\ast }(z)]}^2{g}^s(z,a)dA(z),$$

and

$$I_2(a,r)=\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\le r}{[{(f_{n_j}\circ \varphi )}_p^{\ast }(z)-{(f\circ \varphi )}_p^{\ast }(z)]}^2{g}^s(z,a)dA(z).$$

Since ${(f_{n_j})}_{j=1}^{\mathrm{\infty }}\subset B$ and $f\in B$, it follows that

$$\begin{array}{rcl}{I}_1(a,r)& =& \underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\ge r}{[{(f_{n_j}\circ \varphi )}_p^{\ast }(z)-{(f\circ \varphi )}_p^{\ast }(z)]}^2{g}^s(z,a)dA(z)\\ \le & \frac{p}{2}\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\ge r}\mathcal{L}(f_{n_j},f,\varphi )g^s(z,a)dA(z)\\ \le & d_{{\mathcal{B}}_{\alpha }^{\ast }}(f_{n_j},f)\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\ge r}\frac{{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{g}^s(z,a)dA(z),\end{array}$$

where

$$\mathcal{L}(f_{n_j},f,\varphi )={|\frac{{|(f_{n_j}\circ \varphi )(z)|}^{\frac{p}{2}-1}{(f_{n_j}\circ \varphi )}^{\prime }(z)}{1-{|(f_{n_j}\circ \varphi )(z)|}^p}-\frac{{|(f\circ \varphi )(z)|}^{\frac{p}{2}-1}{(f\circ \varphi )}^{\prime }(z)}{1-{|(f\circ \varphi )(z)|}^p}|}^2.$$

Hence,

$$I_1(a,r)\le C\epsilon .$$

(8)

On the other hand, by the uniform convergence on compact subsets of $\mathbb{D}$, we can find an $N_3\in \mathbb{N}$ such that for all $j\ge N_3$,

$${\mathcal{L}}_1(f_{n_j},f,\varphi )=|\frac{{|(f_{n_j}\circ \varphi )(z)|}^{\frac{p}{2}-1}{f}_{n_j}^{\prime }(\varphi (z))}{1-{|f_{n_j}(\varphi (z))|}^p}-\frac{{|(f\circ \varphi )(z)|}^{\frac{p}{2}-1}{f}^{\prime }(\varphi (z))}{1-{|f(\varphi (z))|}^p}|\le \epsilon$$

for all $z\in \mathbb{D}$ with $|\varphi (z)|\le r$. Hence, for such j, we obtain

$$\begin{array}{rcl}{I}_2(a,r)& =& \underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\le r}{({(f_{n_j}\circ \varphi )}_p^{\ast }(z)-{(f\circ \varphi )}_p^{\ast }(z))}^2{g}^s(z,a)dA(z)\\ \le & \underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\le r}{\mathcal{L}}_1(f_{n_j},f,\varphi ){|{\varphi }^{\mathrm{\prime }}(z)|}^2{g}^s(z,a)dA(z)\\ \le & \epsilon {(\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi (z)|\le r}\frac{{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{g}^s(z,a)dA(z))}^{\frac{1}{2}}\le C\epsilon ,\end{array}$$

hence,

$$I_2(a,r)\le C\epsilon ,$$

(9)

where C is the bound obtained from (iii) of Theorem 3.1. Combining (6), (7), (8) and (9), we deduce that $f_{n_j}\to f$ in $Q^{\ast }(p,s)$.

As for the converse direction, let $f_n(z):=\frac{1}{2}{n}^{\alpha -1}{z}^n$ for all $n\in \mathbb{N}$, $n\ge 2$.

$$\begin{array}{rcl}{\parallel f\parallel }_{{\mathcal{B}}_{p,\alpha }^{\ast }}& =& \frac{p}{2}\underset{a\in \mathbb{D}}{sup}\frac{n^{\frac{\alpha p}{2}}{|z|}^{\frac{\alpha p}{2}-1}{(1-{|z|}^2)}^{\alpha }}{1-2^{-p}{n}^{p(\alpha -1)}{|z|}^{np}}\\ \le & (2^{p-1}+1)\underset{a\in \mathbb{D}}{sup}{n}^{\frac{\alpha p}{2}}{|z|}^{\frac{\alpha p}{2}-1}{(1-{|z|}^2)}^{\alpha }.\end{array}$$

(10)

The function $r^{\frac{np}{2}-1}{(1-r)}^{\alpha }$ attains its maximum at the point $r=1-\frac{\alpha }{\alpha +\frac{\alpha p}{2}-1}$. For simplicity, we see that (10) has the upper bound

$$(2^{p-1}+1)n^{\alpha }{(1-\frac{\alpha }{\alpha +n-1})}^{n-1}{\left(\frac{\alpha }{\alpha +n-1}\right)}^{\alpha }\le (2^{p-1}+1).$$

Then the sequence ${(f_n)}_{n=1}^{\mathrm{\infty }}$ belongs to the ball $\overline{B}(0,(2^{p-1}+1))\subset {\mathcal{B}}_{p,\alpha }^{\ast }$.

Suppose that $C_{\varphi }$ maps the closed ball $\overline{B}(0,(2^{p-1}+1))\subset {\mathcal{B}}_{p,\alpha }^{\ast }$ into a compact subset of $Q^{\ast }(p,s)$; hence, there exists an unbounded increasing subsequence ${(n_j)}_{j=1}^{\mathrm{\infty }}$ such that the image subsequence ${(C_{\varphi }{f}_{n_j})}_{j=1}^{\mathrm{\infty }}$ converges with respect to the norm. Since both ${(f_n)}_{n=1}^{\mathrm{\infty }}$ and ${(C_{\varphi }{f}_{n_j})}_{j=1}^{\mathrm{\infty }}$ converge to the zero function uniformly on compact subsets of $\mathbb{D}$, the limit of the latter sequence must be zero. Hence,

$${\parallel n_j^{\alpha -1}{\varphi }^{n_j}\parallel }_{Q^{\ast }(p,s)}\to 0,\text{as }j\to \mathrm{\infty }.$$

(11)

Now let $r_j=1-\frac{1}{n_j}$. For all numbers a, $r_j\le a<1$, we have the estimate

$$\frac{n_j^{\alpha }{a}^{n_j-1}}{1-a^{n_j}}\ge \frac{1}{e{(1-a)}^{\alpha }}\left(\text{see [14]}\right).$$

(12)

Using (12), we deduce

$$\begin{array}{rcl}{\parallel n_j^{\alpha -1}{\varphi }^{n_j}\parallel }_{Q^{\ast }(p,s)}^2& \ge & \frac{p}{2}\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi |\ge r_j}{\left|\frac{n_j^{\alpha }{(\varphi (z))}^{n_j-1}{|{\varphi }^{n_j}(z)|}^{\frac{p}{2}-1}{\varphi }^{\prime }(z)}{1-{|{\varphi }^{n_j}(z)|}^p}\right|}^2{g}^s(z,a)dA(z)\\ \ge & \frac{Cp}{8e^2}\underset{a\in \mathbb{D}}{sup}{\int }_{|\varphi |\ge r_j}\frac{{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{g}^s(z,a)dA(z).\end{array}$$

(13)

From (11) and (13), the condition (ii) follows. This completes the proof. □

For $0<p<\mathrm{\infty }$ and $0\le s<\mathrm{\infty }$, we define the weighted Dirichlet-class $\mathcal{D}(p,s)$ consists of those functions $f\in \mathcal{H}(\mathbb{D})$ for which

$${\int }_{\mathbb{D}}{|f(z)|}^{p-2}{|f^{\mathrm{\prime }}(z)|}^2{(1-{|z|}^2)}^{s}dA(z)<\mathrm{\infty }.$$

For $0<p<\mathrm{\infty }$ and $0\le s<\mathrm{\infty }$, the generalized hyperbolic weighted Dirichlet-class ${\mathcal{D}}^{\ast }(p,s)$ consists of those functions $f\in B(\mathbb{D})$ for which

$${\int }_{\mathbb{D}}{(f_p^{\ast }(z))}^2{(1-{|z|}^2)}^{s}dA(z)<\mathrm{\infty }.$$

The proof of Proposition 2.2 implies the following corollary:

Corollary 3.1 For$f,g\in {\mathcal{D}}^{\ast }(p,s)$. Then, ${\mathcal{D}}^{\ast }(p,s)$is a complete metric space with respect to the metric defined by

$$d(f,g;{\mathcal{D}}^{\ast }(p,s)):=d_{{\mathcal{D}}^{\ast }(p,s)}(f,g)+{\parallel f-g\parallel }_{\mathcal{D}(p,s)}+{|f(0)-g(0)|}^{\frac{p}{2}},$$

where

$$d_{{\mathcal{D}}^{\ast }(p,s)}(f,g):={(\frac{p}{2}\underset{z\in \mathbb{D}}{sup}{\int }_{\mathbb{D}}{|\frac{f^{\prime }(z){|f(z)|}^{\frac{p}{2}-1}}{1-{|f(z)|}^p}-\frac{g^{\prime }(z){|g(z)|}^{\frac{p}{2}-1}}{1-{|g(z)|}^p}|}^2{(1-{|z|}^2)}^{s}dA(z))}^{\frac{1}{2}}.$$

Moreover, the proofs of Theorems 3.1 and 3.2 yield the following result:

Theorem 3.3 Let$0<p<\mathrm{\infty }$, $-1<s\le 1$, and$0<\alpha \le 1$. Assume that ϕ is a holomorphic mapping from$\mathbb{D}$into itself. Then the following statements are equivalent:

• (i) $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to {\mathcal{D}}^{\ast }(p,s)$ is Lipschitz continuous;

• (ii) $C_{\varphi }:{\mathcal{B}}_{p,\alpha }^{\ast }\to {\mathcal{D}}^{\ast }(p,s)$ is compact;(iii)

  $${\int }_{\mathbb{D}}\frac{{|{\varphi }^{\prime }(z)|}^2}{{(1-{|\varphi (z)|}^p)}^{2\alpha }}{(1-{|z|}^2)}^{s}dA(z)<\mathrm{\infty }.$$