Generalized weighted composition operators from α-Bloch spaces into weighted-type spaces

Abstract

Some criteria for the boundedness, as well as for the compactness, of the generalized weighted composition operator $D^{n}_{\varphi, u}$ from α-Bloch spaces into weighted-type spaces are given. Estimates for the norm and the essential norm of the operator are also given. Our results extend and complement some results in the literature.

1 Introduction

Let $\mathbb{D}$ be the unit disk of the complex plane $\Bbb{C}$, $H(\mathbb{D})$ the class of functions analytic on $\mathbb{D}$, and $H^{\infty}=H^{\infty}(\mathbb{D})$ the space of bounded analytic functions on $\mathbb{D}$. For $0<\alpha <\infty$, an $f\in H(\mathbb{D})$ is said to belong to the α-Bloch space $\mathcal{B}^{\alpha}=\mathcal{B}^{\alpha}(\mathbb{D})$ if

$$b_{\alpha}(f)=\sup_{z \in\mathbb{D}}\bigl(1-|z|^{2} \bigr)^{\alpha}\bigl\vert f'(z)\bigr\vert < \infty. $$

It is easy to check that $\mathcal{B}^{\alpha}$ becomes a Banach space with the norm $\|f\|_{\mathcal{B}^{\alpha}}=|f(0)|+b_{\alpha}(f)$. The little α-Bloch space $\mathcal{B}^{\alpha}_{0}=\mathcal{B}^{\alpha}_{0}(\mathbb{D})$, is a subspace of $\mathcal{B}^{\alpha}$ consisting of all $f\in H(\mathbb{D})$ such that

$$\lim_{|z|\to1^{-}}\bigl(1-|z|^{2}\bigr)^{\alpha}\bigl\vert f'(z)\bigr\vert =0. $$

When $\alpha=1$, $\mathcal{B}^{1}=\mathcal{B}$ is the well-known Bloch space, while $\mathcal{B}^{1}_{0}=\mathcal{B}_{0}$ is the well-known little Bloch space. For some results on the α-Bloch spaces and the little α-Bloch spaces, see, for example, [1].

A positive continuous function on $\mathbb{D}$ is called a weight. Let $\mu(z)$ be a weight. The weighted-type space on $\mathbb{D}$ [2, 3], denoted by $H^{\infty}_{\mu}=H^{\infty}_{\mu}(\mathbb{D})$, consists of all $f\in H(\mathbb{D})$ such that

$$\|f\|_{ H^{\infty}_{\mu}}=\sup_{z\in\mathbb{D}} \mu(z)\bigl\vert f(z)\bigr\vert < \infty. $$

It is obvious that $H^{\infty}_{0}=H^{\infty}$, while for $\mu(z)=(1-|z|^{2})^{\beta}$, $\beta>0$, is obtained the growth space $H^{\infty}_{\beta}$ [4].

Let $u \in H(\mathbb{D})$ and φ be an analytic self-map of $\mathbb{D}$. The weighted composition operator $uC_{\varphi}$, induced by φ and u, is defined by

$$(uC_{\varphi}f) (z) =u(z)\cdot f \bigl(\varphi(z)\bigr) ,\quad f \in H( \mathbb{D}), z\in\mathbb{D}. $$

When $u(z)\equiv1$, then the weighted composition operator is reduced to the composition operator, usually denoted by $C_{\varphi}$, while for $\varphi(z)\equiv z$, it is reduced to the multiplication operator, usually denoted by $M_{u}$.

A natural generalization of the weighted composition operator is the generalized weighted composition operator [5] or the weighted differentiation composition operator [6] $D^{n}_{\varphi, u}$, which is defined as

$$\bigl(D^{n}_{\varphi, u} f\bigr) (z) =u(z)\cdot f^{(n)} \bigl(\varphi(z)\bigr) ,\quad f \in H(\mathbb{D}), z\in\mathbb{D}, $$

where $n\in\mathbb{N}_{0}$, $u \in H(\mathbb{D})$, and φ is an analytic self-map of $\mathbb{D}$. Clearly, when $n=0$ and $u(z)=1$, $D^{n}_{\varphi,u}$ is the composition operator $C_{\varphi}$, if $n=0$, then $D^{n}_{\varphi,u}$ is the weighted composition operator $uC_{\varphi}$. If $n=1$ and $u(z)=\varphi'(z)$, then $D^{n}_{\varphi, u}= DC_{\varphi}$, which was studied, for example, in [3, 7–15], while for $u(z)=1$, $D^{n}_{\varphi, u}= C_{\varphi}D^{n}$, which was studied in [3, 13, 15, 16]. For some other results on the generalized weighted composition operator on various spaces of holomorphic functions, see, for example, [17–22]. A fundamental problem concerning concrete operators is to relate function theoretic properties of their symbols to their operator theoretic properties (see, for example, [3, 5–29]).

It is well known that the composition operator is bounded on the Bloch space $\mathcal{B}$. See, for example, [26, 28, 29] for the compactness and essential norm of the composition operator on $\mathcal{B}$. In [28], it was shown that $C_{\varphi}$ is compact on $\mathcal{B}$ if and only if

$$\|C_{\varphi}p_{j}\|_{\mathcal{B}}=\bigl\Vert \varphi^{j}\bigr\Vert _{\mathcal{B}}\to0 \quad \mbox{as } j\to \infty, $$

where $p_{j}(z)=z^{j}$, $j\in\mathbb{N}_{0}$.

Motivated by this result, in [22], the author proved that $D^{n}_{\varphi,u}: \mathcal{B} \to H^{\infty}_{\beta}$ is compact if and only if it is bounded and

$$\lim_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(p_{j}) \bigr\Vert _{H^{\infty}_{\beta}}=0. $$

Following the line of the above mentioned investigations, in this work, we consider the operators $D^{n}_{\varphi, u} :\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}$, and show that $D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}$ is bounded (respectively, compact) if and only if the sequence $(j^{\alpha -1}\|D^{n}_{\varphi,u} (p_{j})\|_{H_{\mu}^{\infty}})_{j=n}^{\infty}$ is bounded (respectively, convergent to 0 as $j\to\infty$). Moreover, we give some estimates for the norm, as well as for the essential norm of the operator $D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal {B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}$. Recall that the essential norm of the operator $T:X\rightarrow Y$ is its distance to the set of compact operators K mapping X to Y, that is,

$$\|T\|_{e, X\rightarrow Y}=\inf\bigl\{ \Vert T-K\Vert _{X\rightarrow Y}: K \mbox{ is compact}\bigr\} , $$

where X and Y are Banach spaces and $\|\cdot\|_{X\rightarrow Y}$ is the operator norm. Consequently, $\|T\|_{e,X\rightarrow Y}=0$ if and only if T is compact.

Throughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. We write $P\preceq Q$ if there exists a positive constant C independent of the quantities P and Q such that $P\leq CQ$. The symbol $P\approx Q$ means that $P\preceq Q\preceq P$.

2 Boundedness of $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\to H^{\infty}_{\mu}$

For $w\in\mathbb{D}$, set

$$f_{w}(z)=\frac{1-|w|^{2}}{ (1-\overline{w} z)^{\alpha}},\quad z\in\mathbb{D} . $$

Note that

$$ f^{(n)}_{w}(z)=\frac{(1-|w|^{2})\overline{w}^{n}}{ (1-\overline{w} z)^{\alpha +n}}\prod _{j=0}^{n-1}(\alpha +j),\quad z\in \mathbb{D}, n\in \mathbb{N}. $$

(1)

In this section, we will use this family of functions, as well as the sequence of functions $(j^{\alpha -1} p_{j})_{j\in\mathbb{N}}$ to characterize the boundedness and compactness of $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\to H^{\infty}_{\mu}$.

Theorem 2.1

Let n be a positive integer, $\alpha >0$, μ a weight, $u\in H(\mathbb{D})$, and φ be an analytic self-map of $\mathbb{D}$. Then the following statements are equivalent.

• (a) The operator $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}$ is bounded.

• (b) The operator $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}$ is bounded.

• (c) $M_{1}:= \sup_{j \geq n} j^{\alpha -1} \| D^{n}_{\varphi, u}(p_{j})\|_{H^{\infty}_{\mu}}<\infty$.

• (d) $M_{2}:=\sup_{w\in\mathbb{D}}\|D^{n}_{\varphi,u} f_{\varphi (w)}\|_{H^{\infty}_{\mu}}<\infty$ and $u\in H^{\infty}_{\mu}$.

• (e) $M_{3}:= \sup_{z\in\mathbb{D} } \frac{\mu(z)|u(z) |}{(1-|\varphi(z)|^{2})^{ n+\alpha -1}} <\infty$ and $u\in H^{\infty}_{\mu}$.

Moreover, if the operator $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}$ is bounded, then the following asymptotic relations hold:

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{\mathcal{B}^{\alpha}\to H^{\infty}_{\mu}}\approx \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}}\approx M_{1}\approx\max\bigl\{ M_{2},\Vert u\Vert _{H^{\infty}_{\mu}} \bigr\} \approx M_{3}. $$

(2)

Proof

(a) ⇒ (b) Since $\mathcal{B}^{\alpha}_{0}\subset \mathcal{B}^{\alpha}$, this implication, as well as the inequality

$$ \bigl\Vert D^{n}_{\varphi, u} \bigr\Vert _{\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}}\leq \bigl\Vert D^{n}_{\varphi, u} \bigr\Vert _{\mathcal{B}^{\alpha}\to H^{\infty}_{\mu}}, $$

(3)

is obvious.

(b) ⇒ (c) It is easy to see that the sequence $(j^{\alpha -1} p_{j})_{j\in\mathbb{N}}$ is bounded in $\mathcal {B}^{\alpha}_{0}$ and

$$\| p_{j}\|_{\mathcal{B}^{\alpha}}=j \biggl(\frac{2\alpha}{j-1+2\alpha } \biggr)^{\alpha}\biggl(\frac{j-1}{j-1+2\alpha} \biggr)^{\frac {j-1}{2}}, \quad \mbox{for } j\in\mathbb{N}, $$

which implies that $\|j^{\alpha -1}p_{j}\|_{\mathcal{B}^{\alpha}}\approx1$. Notice that $(D^{n}_{\varphi, u} p_{n})(z)=u(z)n!$, $z\in\mathbb{D}$, while for $j< n$, $D^{n}_{\varphi, u} (p_{j}) =0$. Therefore, by the boundedness of $D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}$, we get

$$j^{\alpha -1}\bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} = \bigl\Vert D^{n}_{\varphi, u} \bigl(j^{\alpha -1}p_{j}\bigr)\bigr\Vert _{H^{\infty}_{\mu}} \leq C\bigl\Vert D^{n}_{\varphi, u} \bigr\Vert _{\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}}< \infty, $$

for every $j\in \mathbb{N}$, proving (c), as well as the asymptotic relation

$$ M_{1}\preceq\bigl\Vert D^{n}_{\varphi, u} \bigr\Vert _{\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}}. $$

(4)

(c) ⇒ (a) If $\|\varphi\|_{\infty}=\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1$, then by Proposition 8 in [1], we have

$$\bigl\| D^{n}_{\varphi, u} f\bigr\| _{H_{\mu}^{\infty}}\preceq \frac{\|u\|_{H_{\mu}^{\infty}}\|f\|_{{\mathcal{B}}^{\alpha}}}{(1-\|\varphi\|_{\infty}^{2})^{n+\alpha-1}}, $$

from which the boundedness of $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}$ follows in this case.

Now assume that $\|\varphi\|_{\infty}=1$. Let $\mathbb{D}_{j}=\{z\in \mathbb{D}: r_{j} \leq |\varphi(z)|< r_{j+1}\}$ where $r_{j}=(j-n)/(j+\alpha-1)$ for $j\ge n$. Then from Lemma 1 in [16], which also holds for $m=0$, i.e., $n=1$ in our case, we have that there is a $\delta>0$ such that

$$\min_{z\in \mathbb{D}_{j}}j^{\alpha-1}j(j-1)\cdots(j-n+1)\bigl|\varphi(z)\bigr|^{j-n} \bigl(1-\bigl|\varphi(z)\bigr|\bigr)^{\alpha+n-1}\geq \delta, $$

for every $j\ge k+1$, where k is the smallest natural number such that $\mathbb{D}_{k}\ne\emptyset$.

Fix $N\ge k+1$. Then, clearly $N\ge n+1$ and we have

$$ \bigl\| D^{n}_{\varphi, u}f\bigr\| _{H^{\infty}_{\mu}}\leq \sup_{|\varphi(z)|< \frac{N-n}{N+\alpha-1} }\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr|+ \sup_{|\varphi(z)|\ge \frac{N-n}{N+\alpha-1} }\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr|. $$

(5)

The finiteness of $M_{1}$ implies $u\in H_{\mu}^{\infty}$. Hence, as in the first case, we have

$$ \sup_{|\varphi(z)|< \frac{N-n}{N+\alpha-1} }\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr| \preceq \|u\|_{H^{\infty}_{\mu}}\|f\|_{{\mathcal{B}}^{\alpha}}. $$

(6)

On the other hand, since $\mathbb{D}\setminus \{|\varphi(z)|<\frac{N-n}{N+\alpha-1}\}= \bigcup_{j\ge N}\mathbb{D}_{j}$, we get

$$\begin{aligned}& \sup_{|\varphi(z)|\ge\frac{N-n}{N+\alpha-1} }\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr| \\& \quad =\sup_{j\ge N}\sup_{z\in \mathbb{D}_{j}}\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr| \\& \quad =\sup_{j\ge N}\sup_{z\in \mathbb{D}_{j}}\mu(z)\bigl|u(z)\bigr| \frac{j^{\alpha-1}j(j-1)\cdots(j-n+1) |\varphi(z)|^{j-n} |f^{(n)}(\varphi(z))|(1-|\varphi(z)|)^{\alpha+n-1}}{j^{\alpha-1}j(j-1)\cdots(j-n+1)(1-|\varphi(z)|)^{\alpha+n-1}|\varphi(z)|^{j-n}} \\& \quad \preceq\frac{\|f\|_{{\mathcal{B}}^{\alpha}}}{\delta} \sup_{j\ge N} j^{\alpha-1} \bigl\| D^{n}_{\varphi, u}(p_{j})\bigr\| _{H^{\infty}_{\mu}} \leq \frac{M_{1}}{\delta}\|f\|_{{\mathcal{B}}^{\alpha}}< \infty. \end{aligned}$$

(7)

From (5), (6) and (7), the boundedness of $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}$ follows.

(c) ⇒ (d) First note that (c) implies that $u\in H^{\infty}_{\mu}$. Further, since

$$\sup_{z\in{\mathbb{D}}}\bigl(1-|z|^{2}\bigr)^{\alpha}\bigl|f'_{w}(z)\bigr|=\sup_{z\in {\mathbb{D}}} \bigl(1-|z|^{2}\bigr)^{\alpha}\frac{|\alpha \overline{w}|(1-|w|^{2})}{|1-\overline{w} z|^{\alpha +1}}\leq |\alpha |2^{\alpha +1}, \quad w\in{\mathbb{D}}, $$

the family of functions $(f_{w})_{w\in\mathbb{D}}$ is uniformly bounded in $\mathcal{B}^{\alpha}$. Furthermore

$$f_{w}(z)=\bigl(1-|w|^{2}\bigr) \sum _{j=0}^{\infty}\frac{\Gamma(j+\alpha)}{j!\Gamma(\alpha )}\overline{w}^{j}z^{j},\quad z \in \mathbb{D}. $$

By Stirling’s formula, we have $\frac{\Gamma(j+\alpha)}{j!\Gamma(\alpha)}\approx j^{\alpha-1}$ as $j\rightarrow\infty$. Using this fact, the linearity and continuity of the operator, we get

$$\bigl\Vert D^{n}_{\varphi, u} f_{w}\bigr\Vert _{H^{\infty}_{\mu}} \leq C\bigl(1-|w|^{2}\bigr)\sum _{j=n}^{\infty}|w|^{j}j^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u} (p_{j})\bigr\Vert _{H^{\infty}_{\mu}}\preceq M_{1}< \infty,\quad w\in\mathbb{D}. $$

Consequently, $\sup_{w\in\mathbb{D}} \|D^{n}_{\varphi, u} f_{\varphi(w)}\|_{H^{\infty}_{\mu}}\preceq M_{1}$, and along with the inequality $n^{\alpha -1}n!\|u\|_{H^{\infty}_{\mu}}\leq M_{1}$, obtained by considering $\|D^{n}_{\varphi, u}(n^{\alpha -1}p_{n})\|_{H^{\infty}_{\mu}}$, we also have

$$ \max\bigl\{ M_{2},\Vert u\Vert _{H^{\infty}_{\mu}}\bigr\} \preceq M_{1}. $$

(8)

(d) ⇒ (e) For $\lambda\in\mathbb{D}$, it follows from (d) and (1) that

$$\begin{aligned} M_{2} \geq \bigl\Vert D^{n}_{\varphi,u} f_{\varphi(\lambda)}\bigr\Vert _{H^{\infty}_{\mu}} \geq\frac{\mu(\lambda)|u(\lambda)| |\varphi(\lambda)|^{n}\prod_{j=0}^{n-1}(\alpha +j)}{(1-|\varphi(\lambda)|^{2})^{n+\alpha -1}}. \end{aligned}$$

(9)

For any fixed $r\in(0,1)$, from (9), we have

$$ \sup_{|\varphi(\lambda)|>r} \frac{\mu(\lambda)|u(\lambda)| }{(1-|\varphi(\lambda)|^{2})^{ n+\alpha -1}} \leq \sup_{|\varphi(\lambda)|>r} \frac{ |\varphi(\lambda)|^{n }}{r^{n }} \frac{\mu(\lambda) |u(\lambda)| }{(1-|\varphi(\lambda)|^{2})^{ n+\alpha -1}} \preceq\frac{M_{2}}{r^{n }}. $$

(10)

On the other hand, from $u \in H^{\infty}_{\mu}$, we have

$$\begin{aligned} \sup_{|\varphi(\lambda)|\leq r}\frac{\mu(\lambda)|u(\lambda)| }{(1-|\varphi(\lambda)|^{2})^{ n+\alpha -1}} & \leq\frac{\sup_{|\varphi(\lambda)|\leq r} \mu (\lambda) |u(\lambda)|}{(1-r^{2})^{ n+\alpha -1}} \\ &\leq \frac{\|u\|_{H^{\infty}_{\mu}}}{(1-r^{2})^{ n+\alpha -1}}< \infty. \end{aligned}$$

(11)

Therefore, (10) and (11) yield the inequality of (e), as well as the asymptotic relation

$$ M_{3}\preceq \max\bigl\{ M_{2},\Vert u\Vert _{H^{\infty}_{\mu}}\bigr\} . $$

(12)

(e) ⇒ (a) By Proposition 8 in [1], if $f \in \mathcal{B}^{\alpha}$ and $k\in\mathbb{N}$, we see that

$$\sup_{z\in\mathbb{D}}\bigl(1-|z|^{2}\bigr)^{k+\alpha -1}\bigl\vert f^{(k)}(z)\bigr\vert \leq C\|f\| _{\mathcal {B}^{\alpha}}, $$

for some constant C independent of f. Therefore, for $z\in\mathbb{D}$, we have

$$\begin{aligned} \mu(z)\bigl\vert \bigl(D^{n}_{\varphi,u} f\bigr) (z) \bigr\vert =&\mu(z)\bigl\vert u(z)\bigr\vert \bigl\vert f^{(n)}\bigl( \varphi(z)\bigr)\bigr\vert \\ \leq& C\frac{\mu(z)|u(z)| }{(1-|\varphi(z)|^{2})^{n+\alpha -1}}\|f\|_{\mathcal{B}^{\alpha}}, \end{aligned}$$

(13)

where C is independent of f. Taking the supremum in (13) over $\mathbb{D}$ and then using the first condition in (e) we see that $D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}$ is bounded, and

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{\mathcal{B}^{\alpha}\to H^{\infty}_{\mu}}\preceq M_{3}. $$

(14)

If the operator $D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}$ is bounded, then from (3), (4), (8), (12), and (14), we obtain (2), completing the proof. □

3 Compactness and essential norm of $D^{n}_{\varphi,u}: \mathcal {B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\rightarrow H^{\infty}_{\mu}$

In this section we will give an estimate for the essential norm of the operator $D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}$, as well as of $D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}$. For this purpose, we state several lemmas, which will be used in the proof of the main result.

Lemma 3.1

[16]

Let $\alpha >0$, $m\geq n+1$, where $n\in\mathbb{N}$. Define the function $H_{m,\alpha }:[0,1]\rightarrow[0,\infty)$ by

$$ H_{m,\alpha}(x)=\frac{m!}{(m-n-1)!}x^{m-n-1}(1-x)^{n+\alpha }. $$

Then the following statements hold:

• (i)
  $$\max_{0\leq x\leq1}H_{m, \alpha}(x)=H_{m,\alpha}(r_{m})= \left \{ \textstyle\begin{array}{l@{\quad}l} (n+1)!, & m=n+1, \\ \frac{m!}{(m-n-1)!} (\frac{m-n-1}{m+\alpha-1} )^{m-n-1} (\frac{n+\alpha}{m+\alpha-1} )^{\alpha+n} , & m > n+1 , \end{array}\displaystyle \right . $$

  where

  $$r_{m}= \left \{ \textstyle\begin{array}{l@{\quad}l} 0, & m=n+1, \\ \frac{m-n-1}{m+\alpha-1} , & m > n+1 . \end{array}\displaystyle \right . $$

• (ii) For $m>n+1$, $H_{m,\alpha }$ is decreasing on $[r_{m}, r_{m+1}]$, and so

  $$ \min_{r_{m}\leq x\leq r_{m+1}}H_{m, \alpha}(x)=H_{m,\alpha}(r_{m+1})= \frac{m!}{(m-n-1)!} \biggl(\frac {m-n}{m+\alpha} \biggr)^{m-n-1} \biggl( \frac{n+\alpha}{m+\alpha} \biggr)^{\alpha+n} . $$

  Consequently,

  $$ \lim_{m\rightarrow\infty} m^{\alpha-1} \min_{r_{m}\leq x\leq r_{m+1}}H_{m, \alpha}(x) = \frac{(n+\alpha)^{n+\alpha}}{e^{n+\alpha}}. $$

Denote by $K_{r}f(z)=f(rz)$ for $r\in(0,1)$ and $z\in\mathbb{D}$. Then $K_{r}$ is a compact operator on ${\mathcal{B}}^{\alpha}$ for every $\alpha >0$, and $\|K_{r}\|\leq1$ (see, e.g., Proposition 1.3 in [24] and [27]). Let I denote the identity operator. The following three lemmas can be found in [25] (see also [16]).

Lemma 3.2

Let $0<\alpha<1$. Then there is a sequence $(r_{k})_{k\in\mathbb{N}}$, with $0< r_{k}<1$ tending to 1, such that the sequence of compact operators $L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}$, $j\in\mathbb{N}$, on $\mathcal {B}_{0}^{\alpha}$ satisfies the following.

• (i) For any $t\in(0,1)$, $\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0$.

• (ii) $\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}^{\alpha}}\leq1}\sup_{z\in\mathbb{D}}|(I-L_{j})f(z)|=0$.

• (iii) $\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1$.

Furthermore, these statements hold as well for the sequence of biadjoints $L^{**}_{j}$ on $\mathcal{B}^{\alpha}$.

Lemma 3.3

Let $\alpha=1$. Then there is a sequence $(r_{k})_{k\in\mathbb{N}}$, with $0< r_{k}<1$ tending to 1, such that the sequence of compact operators $L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}$, $j\in\mathbb{N}$, on $\mathcal{B}_{0}$ satisfies the following.

• (i) For any $t\in[0,1)$, $\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0$.

• (iia) $\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}}\leq1}\sup_{|z|>s}|(I-L_{j})f(z)| (\log\frac {1}{1-|z|^{2}} )^{-1}\leq1$, for s sufficiently close to 1.

• (iib) $\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}}\leq1}\sup_{|z|\leq s}|(I-L_{j})f(z)|=0$ for the above s.

• (iii) $\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1$.

Furthermore, these statements hold as well for the sequence of biadjoints $L^{**}_{j}$ on $\mathcal{B}$.

Lemma 3.4

Let $\alpha>1$. Then there is a sequence $(r_{k})_{k\in\mathbb{N}}$, with $0< r_{k}<1$ tending to 1, such that the sequence of compact operators $L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}$, $j\in\mathbb{N}$, on $\mathcal {B}_{0}^{\alpha}$ satisfies the following.

• (i) For any $t\in[0,1)$, $\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0$.

• (ii) For any $t\in[0,1)$, $\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|(I-L_{j})f(z)|=0$.

• (iii) $\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1$.

Furthermore, these statements hold as well for the sequence of biadjoints $L^{**}_{j}$ on $\mathcal{B}^{\alpha}$.

To study the compactness, we also need the following lemma, which can be proved in a standard way (see, for example, Proposition 3.11 in [23]).

Lemma 3.5

Let n be a nonnegative integer, $\alpha >0$, μ a weight, $u \in H(\mathbb{D})$ and φ be an analytic self-map of $\mathbb{D}$. Then $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\ (\textit{or } \mathcal{B}^{\alpha}_{0})\rightarrow H^{\infty}_{\mu}$ is compact if and only if $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\ (\textit{or } \mathcal {B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}$ is bounded and for any bounded sequence $(f_{k})_{k\in{\mathbb{N}}}$ in $\mathcal{B}^{\alpha}$, which converges to zero uniformly on compact subsets of $\mathbb{D}$,

$$\lim_{k\to\infty}\bigl\Vert D^{n}_{\varphi,u} f_{k}\bigr\Vert _{H^{\infty}_{\mu}}=0. $$

Now we are ready to state and prove the main results in this section.

Theorem 3.6

Let n be a positive integer, $\alpha >0$, μ a weight, $u\in H(\mathbb{D})$, and φ be an analytic self-map of $\mathbb{D}$. Suppose that $D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}$ is bounded. Then

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}\approx \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}\approx \limsup_{j\rightarrow\infty}j^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} . $$

(15)

Proof

First note that the inequality

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}\leq \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} $$

(16)

obviously holds.

Now we give a lower estimate for the essential norm $\|D^{n}_{\varphi, u}\|_{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}$. Without loss of generality, we assume that $j\geq n$. Choose the sequence of functions $q_{j}=j^{\alpha-1} p_{j}\in\mathcal{B}^{\alpha}_{0}$, $j\in\mathbb{N}$. Then $\|q_{j}\|_{\mathcal{B}^{\alpha}}\approx1$, and $(q_{j})_{j\in\mathbb{N}}$ converges to zero weakly on $\mathcal{B}^{\alpha}_{0}$ as $j\rightarrow\infty$ (see, for example, Theorem 7.5 in [30]). Since by a well-known theorem, for any compact operator $\widehat{K}:X\to Y$, where X and Y are Banach spaces, the weak convergence $x_{n}\stackrel{w}{\to}x_{0}$ implies the norm convergence $\widehat{K}x_{n}\to\widehat{K}x_{0}$ [31], we have

$$ \lim_{j\rightarrow\infty}\|Kq_{j}\|_{H^{\infty}_{\mu}}=0, $$

(17)

for any given compact operator K from $\mathcal{B}^{\alpha}_{0}$ to $H^{\infty}_{\mu}$.

Hence

$$\bigl\Vert D^{n}_{\varphi, u}-K\bigr\Vert _{\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}} \succeq\bigl\Vert \bigl(D^{n}_{\varphi, u}-K\bigr)q_{j} \bigr\Vert _{H^{\infty}_{\mu}}\geq\bigl\Vert D^{n}_{\varphi, u} q_{j}\bigr\Vert _{H^{\infty}_{\mu}}-\Vert Kq_{j}\Vert _{H^{\infty}_{\mu}}. $$

Letting $j\to\infty$ in the last relation and using (17), we obtain

$$\bigl\Vert D^{n}_{\varphi, u}-K\bigr\Vert _{\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}} \succeq\limsup_{j\rightarrow\infty}\bigl\Vert D^{n}_{\varphi, u} q_{j}\bigr\Vert _{H^{\infty}_{\mu}}=\limsup_{j\rightarrow\infty}j^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{j})\bigr\Vert _{H^{\infty}_{\mu}}, $$

and consequently

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}=\inf _{K} \bigl\Vert D^{n}_{\varphi, u}-K\bigr\Vert _{\mathcal{B}^{\alpha}_{0} \rightarrow H^{\infty}_{\mu}}\succeq\limsup_{j\rightarrow\infty}j^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{j})\bigr\Vert _{H^{\infty}_{\mu}} . $$

(18)

Now, we give the upper estimates for the essential norm $\|D^{n}_{\varphi, u}\|_{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}$. For the case of $\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1$, there is a number $\delta\in(0,1)$ such that $\sup_{z\in{\mathbb{D}}}|\varphi (z)|<\delta$. In this case, the operator $D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}$ is compact. Indeed, choose a bounded sequence $(f_{j})_{j\in\mathbb{N}}$ in $\mathcal {B}^{\alpha}$ which converges to zero uniformly on compact subsets of ${\mathbb {D}}$. From Cauchy’s integral formula, $(f^{(n)}_{j})_{j\in\mathbb{N}}$ also converges to zero on compact subsets of ${\mathbb{D}}$ as $j\rightarrow\infty$. Hence

$$\begin{aligned} \lim_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u} f_{j}\bigr\Vert _{H^{\infty}_{\mu}} =& \lim_{j\rightarrow\infty} \sup _{z\in{\mathbb{D}}}\mu(z) \bigl\vert u(z)f_{j}^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \\ \leq&\Vert u\Vert _{H^{\infty}_{\mu}} \lim_{j\rightarrow \infty} \sup _{z\in{\mathbb{D}}}\bigl\vert f_{j}^{(n)}\bigl(\varphi(z) \bigr)\bigr\vert \\ =&\Vert u\Vert _{H^{\infty}_{\mu}}\lim_{j\rightarrow\infty} \sup _{|w|\leq \delta}\bigl\vert f_{j}^{(n)}(w)\bigr\vert =0. \end{aligned}$$

From this and by Lemma 3.5 we see that the operator $D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}$ is compact. This also shows that

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}=0. $$

(19)

From (16), (18), and (19), we get the desired result in the case $\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1$.

Next, we assume that $\sup_{z\in{\mathbb{D}}}|\varphi(z)|=1$. Let $(L_{j})_{j\in\mathbb{N}}$ be the sequence of operators given in Lemmas 3.2-3.4. Since $L_{j}^{**}$ is compact on $\mathcal{B}^{\alpha}$, for every $j\in\mathbb{N}$, and $D^{n}_{\varphi, u}$ is bounded from $\mathcal {B}^{\alpha}$ to $H^{\infty}_{\mu}$, then $D^{n}_{\varphi, u} L_{j}^{**}$ is also compact from $\mathcal{B}^{\alpha}$ to $H^{\infty}_{\mu}$. Hence

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} \leq& \limsup_{j\rightarrow\infty}\bigl\Vert D^{n}_{\varphi, u} -D^{n}_{\varphi, u} L_{j}^{**}\bigr\Vert _{\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} \\ =&\limsup_{j\rightarrow\infty}\bigl\Vert D^{n}_{\varphi, u} \bigl(I-L_{j}^{**}\bigr)\bigr\Vert _{\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} \\ =&\limsup_{j\rightarrow\infty}\sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq 1}\bigl\Vert D^{n}_{\varphi, u}\bigl(I-L_{j}^{**}\bigr)f\bigr\Vert _{H^{\infty}_{\mu}} \\ =&\limsup_{j\rightarrow\infty}\sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq 1}\sup _{z\in{\mathbb{D}}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert . \end{aligned}$$

For each positive integer $i\geq n$, we define ${\mathbb{D}}_{i}=\{z\in {\mathbb{D}}: r_{i}\leq|\varphi(z)|< r_{i+1}\}$, where $r_{i}$ is given in Lemma 3.1. Let k be the smallest positive integer such that ${\mathbb{D}}_{k}\neq\emptyset$. Since $\sup_{z\in{\mathbb {D}}}|\varphi(z)|=1$, ${\mathbb{D}}_{i}$ is not empty for every integer $i\geq k$ and $\mathbb{D}=\bigcup^{\infty}_{i=k}{\mathbb{D}}_{i}$, we have

$$\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq 1}\sup_{z\in{\mathbb{D}}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**}\bigr)f\bigr)^{(n)}\bigl( \varphi(z)\bigr)\bigr\vert =I_{1}+I_{2}, $$

where

$$I_{1}=\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{k\leq i\leq N-1}\sup _{z\in{\mathbb{D}}_{i}}\mu (z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert $$

and

$$I_{2}=\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{N\leq i}\sup _{z\in {\mathbb{D}}_{i}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert . $$

Here N is a positive integer determined as follows.

By Lemma 3.1, $\lim_{i\rightarrow\infty} \frac{i^{1-\alpha }}{H_{i,\alpha}(r_{i+1})}= \frac{e^{n+\alpha}}{(n+\alpha)^{n+\alpha}}$. Hence, for any given $\varepsilon>0$, there exists an $N\in\mathbb{N}$ such that

$$\frac{i^{ 1-\alpha}}{H_{ i,\alpha}(r_{i+1})}\leq \frac{e^{n+\alpha}}{(n+\alpha)^{n+\alpha}} +\varepsilon $$

when $i\geq N$. For such N it follows that

$$\begin{aligned} I_{2} =&\sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq1}\sup_{N\leq i} \sup_{z\in{\mathbb{D}}_{i}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \\ =& \sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq1}\sup_{N\leq i}\sup _{z\in {\mathbb{D}}_{i}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \frac{H_{ i,\alpha}(|\varphi(z)|)}{i^{1-\alpha}} \frac{i^{1-\alpha } }{H_{ i,\alpha}(|\varphi(z)|)} \\ \preceq& \biggl(\frac{e^{n+\alpha}}{(n+\alpha)^{n+\alpha }}+\varepsilon \biggr)\sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq 1} \bigl\Vert \bigl(I-L_{j}^{**}\bigr)f\bigr\Vert _{\mathcal{B}^{\alpha}}\sup_{N\leq i}\sup_{z\in {\mathbb{D}}_{i}} \mu(z) \bigl\vert u(z)\bigr\vert \frac{i!}{(i-n)!}\frac{|\varphi(z)|^{i-n}}{i^{ 1-\alpha }} \\ \preceq&\bigl\Vert I-L_{j}^{**}\bigr\Vert \sup _{N\leq i}i^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{i}) \bigr\Vert _{H^{\infty}_{\mu}} . \end{aligned}$$

Thus

$$ \limsup_{j\rightarrow\infty}I_{2}\preceq\sup_{i\geq N}i^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{i})\bigr\Vert _{H^{\infty}_{\mu}} . $$

(20)

By Lemmas 3.2, 3.3, 3.4, and Cauchy’s integral formula, we have

$$\begin{aligned} \limsup_{j\rightarrow\infty}I_{1} =& \limsup _{j\rightarrow\infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq 1}\sup_{k\leq i\leq N-1} \sup_{z\in{\mathbb{D}}_{i}}\mu (z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \\ \leq&\|u\|_{H^{\infty}_{\mu}}\limsup_{j\rightarrow\infty}\sup _{\|f\| _{\mathcal{B}^{\alpha}}\leq 1}\sup_{|\varphi(z)|< r_{N}}\bigl\vert \bigl( \bigl(I-L_{j}^{**}\bigr)f\bigr)^{(n)}\bigl(\varphi(z) \bigr)\bigr\vert =0, \end{aligned}$$

which together with (20) implies that

$$\begin{aligned}& \limsup_{j\rightarrow\infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq 1}\sup _{z\in{\mathbb{D}}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi (z)\bigr)\bigr\vert \\& \quad \preceq \sup _{i\geq N}i^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{i}) \bigr\Vert _{H^{\infty}_{\mu}}. \end{aligned}$$

Therefore

$$\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} \preceq \sup_{i\geq N} i^{\alpha-1}\bigl\Vert D^{n}_{\varphi,u}(p_{i}) \bigr\Vert _{H^{\infty}_{\mu}}. $$

From the last relation we get

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}\preceq \limsup_{i\rightarrow\infty} i^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{i})\bigr\Vert _{H^{\infty}_{\mu}}. $$

(21)

From (16), (18), and (21), the asymptotic relations in (15) follow, completing the proof of the theorem. □

From Theorem 3.6, letting $\alpha =1$ we deduce the following result.

Corollary 3.7

Let n be a positive integer, μ a weight, $u \in H(\mathbb{D})$, and φ be an analytic self-map of $\mathbb{D}$. Suppose that $D^{n}_{\varphi, u}:\mathcal{B} \rightarrow H^{\infty}_{\mu}$ is bounded. Then

$$\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B} \rightarrow H^{\infty}_{\mu}}\approx \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}_{0} \rightarrow H^{\infty}_{\mu}}\approx \limsup_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} . $$

Theorem 3.8

Let n be a positive integer, $\alpha >0$, μ a weight, $u\in H(\mathbb{D})$ and φ be an analytic self-map of $\mathbb{D}$. If $D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}$ is bounded, then the following statements are equivalent.

• (a) The operator $D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}$ is compact.

• (b) The operator $D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}$ is compact.

• (c) $\lim_{j\rightarrow\infty} j^{\alpha -1}\|D^{n}_{\varphi, u} (p_{j})\|_{H^{\infty}_{\mu}}=0$.

• (d) $\lim_{|\varphi(w)|\rightarrow1} \|D^{n}_{\varphi,u} f_{\varphi (w)}\|_{H^{\infty}_{\mu}}=0$.

• (e) $\lim_{|\varphi(z)|\rightarrow 1}\frac{\mu(z)|u(z)| }{(1-|\varphi(z)|^{2})^{n+\alpha -1}}=0$.

Proof

The equivalence of statements (a)-(c) follows from Theorem 3.6.

(c) ⇒ (d) From (c), we see that, for every $\varepsilon>0$, there is an $N\in\mathbb{N}$ such that

$$j^{\alpha -1}\bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}}< \varepsilon/2, $$

for all $j\geq N$.

Let $(z_{k})_{k\in\mathbb{N}}\subset{\mathbb{D}}$ be an arbitrary sequence such that $|\varphi (z_{k})|\to1$ as $k\to\infty$ (if such a sequence does not exist then the equality in (d) vacuously holds). Similarly to the proof of Theorem 2.1, we have

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}f_{\varphi (z_{k})}\bigr\Vert _{H^{\infty}_{\mu}} \leq & C\bigl(1-\bigl\vert \varphi (z_{k})\bigr\vert ^{2}\bigr)\sum_{j=n}^{\infty}\bigl\vert \varphi (z_{k})\bigr\vert ^{j}j^{\alpha -1} \bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} \\ =&C\bigl(1-\bigl\vert \varphi (z_{k})\bigr\vert ^{2} \bigr)\sum_{j=n}^{N-1} \bigl\vert \varphi (z_{k})\bigr\vert ^{j}j^{\alpha -1}\bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} \\ &{}+C\bigl(1-\bigl\vert \varphi (z_{k})\bigr\vert ^{2} \bigr)\sum_{j=N}^{\infty} \bigl\vert \varphi (z_{k})\bigr\vert ^{j}j^{\alpha -1}\bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} \\ \leq & 2C\bigl(1-\bigl\vert \varphi (z_{k})\bigr\vert ^{N}\bigr)M_{0}+C\varepsilon, \end{aligned}$$

(22)

for $k\in\mathbb{N}$, where $M_{0}=\max_{n\leq j \leq N-1}j^{\alpha -1}\|D^{n}_{\varphi, u} (p_{j}) \|_{H^{\infty}_{\mu}}$.

Since $|\varphi (z_{k})|\to1$ as $k\to\infty$, from (22), we deduce that

$$\limsup_{k\to\infty}\bigl\Vert D^{n}_{\varphi,u}f_{\varphi (z_{k})} \bigr\Vert _{H^{\infty}_{\mu}} \leq C\varepsilon. $$

Since ε is an arbitrary positive number, the implication follows.

(d) ⇒ (e) Let $(z_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{D}$ such that $\lim_{k\to\infty}|\varphi (z_{k})|=1$ (if such a sequence does not exist then the implication vacuously holds). Since the sequence $(f_{\varphi (z_{k})})_{k\in\mathbb{N}}$ is bounded in $\mathcal{B}^{\alpha}$ and converges to 0 uniformly on compact subsets of $\mathbb{D}$, by (9) and Lemma 3.5, we have

$$\frac{\mu(z_{k}) \vert u(z_{k})\vert |\varphi(z_{k})|^{n}\prod_{j=0}^{n-1}(\alpha +j)}{(1-|\varphi (z_{k})|^{2})^{n+\alpha -1}}\leq \bigl\Vert D^{n}_{\varphi,u} f_{\varphi (z_{k})} \bigr\Vert _{H^{\infty}_{\mu}} \rightarrow0\quad \mbox{as } k \rightarrow\infty. $$

Therefore

$$ \lim_{k\to\infty}\frac{\mu(z_{k}) |u(z_{k})| }{(1-|\varphi(z_{k})|^{2})^{n+\alpha -1}}= \lim_{k\to\infty} \frac{\mu(z_{k}) |u(z_{k})| |\varphi(z_{k})|^{n } }{(1-|\varphi(z_{k})|^{2})^{n+\alpha -1}}=0, $$

(23)

which implies (e).

(e) ⇒ (a) Assume $(f_{k})_{k\in\mathbb{N}}$ is a bounded sequence in $\mathcal{B}^{\alpha}$ converging to 0 uniformly on compact subsets of $\mathbb{D}$. By the assumption, for any $\varepsilon>0$, there exists a $\delta\in(0,1)$ such that

$$ \frac{\mu(z)|u(z)|}{(1-|\varphi(z)|^{2})^{n+\alpha -1}}< \varepsilon $$

(24)

when $\delta<|\varphi(z)|<1$.

Therefore, since $u\in H^{\infty}_{\mu}$ we have

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}f_{k}\bigr\Vert _{H^{\infty}_{\mu}} =& \sup_{z\in\mathbb {D}} \mu(z)\bigl\vert \bigl(D^{n}_{\varphi,u}f_{k}\bigr) (z)\bigr\vert \\ \leq& \sup_{z\in\Omega_{\delta}}\mu(z)\bigl\vert u(z)\bigr\vert \bigl\vert f_{k}^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \\ &{}+C\sup _{z\in\mathbb{D}\setminus\Omega_{\delta}} \frac{\mu(z)|u(z)| }{(1-|\varphi(z)|^{2})^{n+\alpha -1}} \Vert f_{k}\Vert _{\mathcal{B}^{\alpha}} \\ \leq& \Vert u\Vert _{H^{\infty}_{\mu}}\sup_{z\in\Omega_{\delta}} \bigl\vert f_{k}^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert +C \varepsilon \Vert f_{k}\Vert _{\mathcal{B}^{\alpha}}, \end{aligned}$$

(25)

where $\Omega_{\delta}=\{ z\in\mathbb{D}:|\varphi(z)| \leq\delta\}$.

Since $(f_{k})_{k\in\mathbb{N}}$ converges to 0 uniformly on compact subsets of $\mathbb{D}$, by Cauchy’s estimate so do the sequences $(f^{(n)}_{k})_{k\in\mathbb{N}}$ for every $n\in\mathbb{N}$. Letting $k\to \infty$ in (25) and using the fact that ε is an arbitrary positive number, we obtain $\lim_{k\to\infty}\|D^{n}_{\varphi,u} f_{k}\|_{H^{\infty}_{\mu}}=0$. By Lemma 3.5, we deduce that $D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\to H_{\mu}^{\infty}$ is compact. □