Essential norm of generalized weighted composition operators from the Bloch space to the Zygmund space

Abstract

In this paper, we give some estimates of the essential norm for generalized weighted composition operators from the Bloch space to the Zygmund space. Moreover, we give a new characterization for the boundedness and compactness of the operator.

1 Introduction

Let X and Y be Banach spaces. The essential norm of a bounded linear operator $T:X\rightarrow Y$ is its distance to the set of compact operators K mapping X into Y, that is,

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

where $\|\cdot\|_{X\rightarrow Y}$ is the operator norm.

Let $\mathbb{D}$ be the open unit disk in the complex plane $\mathbb{C}$ and $H(\mathbb{D})$ the space of analytic functions on $\mathbb{D}$. Let φ be a nonconstant analytic self-map of $\mathbb{D}$, $u \in H(\mathbb{D})$, and n be a nonnegative integer. The generalized weighted composition operator, denoted by $D^{n}_{\varphi, u}$, is defined on $H(\mathbb{D})$ by

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

When $n=0$, the generalized weighted composition operator $D^{n}_{\varphi , u}$ is the weighted composition operator, denoted by $uC_{\varphi}$. In particular, when $n=0$ and $u=1$, we get the composition operator $C_{\varphi}$. If $n=1$ and $u(z)=\varphi'(z)$, then $D^{n}_{\varphi, u}= DC_{\varphi}$, which was widely studied, for example, in [1–9]. If $u(z)=1$, then $D^{n}_{\varphi, u}= C_{\varphi}D^{n}$, which was studied, for example, in [1, 5, 10, 11]. For the study of the generalized weighted composition operator on various function spaces see, for example, [12–21]. Recently there has been a huge interest in the study of various related product-type operators containing composition operators; see, e.g., [22–30] and the references therein.

The Bloch space, denoted by $\mathcal{B}$, is defined to be the set of all $f \in H(\mathbb{D})$ such that

$$\|f\|_{\mathcal{B}}=\bigl\vert f(0)\bigr\vert +\sup_{z \in\mathbb{D}} \bigl(1-|z|^{2}\bigr) \bigl\vert f'(z)\bigr\vert < \infty. $$

$\mathcal{B}$ is a Banach space with the above norm. An $f\in \mathcal{B}$ is said to belong to the little Bloch space $\mathcal {B}_{0}$ if $\lim_{|z|\rightarrow1}|f'(z)|(1-|z|^{2})=0$. See [31] for more information of Bloch spaces. Composition operators, as well as weighted composition operators mapping into Bloch-type spaces were studied a lot see, for example, [3, 6, 16, 32–45].

The Zygmund space, denoted by $\mathcal{Z}$, is the space consisting of all $f \in H(\mathbb{D})$ such that

$$\|f\|_{\mathcal{Z}}=\bigl\vert f(0)\bigr\vert +\bigl\vert f'(0)\bigr\vert +\sup_{z \in\mathbb{D}}\bigl(1-\vert z \vert ^{2}\bigr) \bigl\vert f''(z)\bigr\vert < \infty. $$

It is easy to see that $\mathcal{Z}$ is a Banach space with the above norm $\| \cdot\|_{\mathcal{Z}}$. See [4, 7, 12, 15, 16, 22, 36, 46–50] for some results of the Zygmund space and related operators mapping into the Zygmund space or into some of its generalizations.

In 1995, Madigan and Matheson proved that $C_{\varphi}:\mathcal {B}\rightarrow \mathcal{B}$ is compact if and only if (see [38])

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

In 1999, Montes-Rodrieguez in [40] obtained the exact value for the essential norm of the operator $C_{\varphi}: \mathcal {B}\rightarrow\mathcal{B}$, i.e.,

$$\|C_{\varphi}\|_{e, \mathcal{B}\rightarrow\mathcal{B}}=\lim_{s\rightarrow1}\sup _{|\varphi(z)|>s}\frac{(1-|z |^{2}) |\varphi'(z)|}{(1-|\varphi(z)|^{2}) }. $$

Tjani in [43] proved that $C_{\varphi}:\mathcal{B}\rightarrow\mathcal{B}$ is compact if and only if $\lim_{|a|\rightarrow1} \| C_{\varphi}\sigma_{a} \| _{\mathcal{B}}=0$, where $\sigma_{a} = \frac{a-z}{1-\overline{a}z}$. Wulan et al. in [44] showed that $C_{\varphi}:\mathcal{B}\rightarrow\mathcal{B}$ is compact if and only if $\lim_{j\rightarrow\infty}\| \varphi^{j} \|_{\mathcal{B}}=0$. Ohno et al. studied the boundedness and compactness of the operator $u C_{\varphi}$ on the Bloch space in [41]. The estimate for the essential norm of the operator $u C_{\varphi}$ on the Bloch space was given in [37]. Some new estimates for the essential norm of $u C_{\varphi}$ on the Bloch space were given in [33, 39]. In [21], Zhu has obtained some estimates for the essential norm of $D^{n}_{\varphi,u}$ on the Bloch space when n is a positive integer.

Stević studied the boundedness and compactness of $D^{n}_{\varphi,u}: \mathcal{B} \rightarrow \mathcal{Z}$ in [16] (see also [50]). In [12], Li and Fu obtained a new characterization for the boundedness, as well as the compactness for $D^{n}_{\varphi,u}: \mathcal{B} \rightarrow\mathcal{Z}$ by using three families of functions. We combine the results in [12] and [16] as follows.

Theorem A

Let n be a positive integer, $u \in H(\mathbb{D})$, and φ be an analytic self-map of $\mathbb{D}$. Suppose that $D^{n}_{\varphi,u} : \mathcal {B}\rightarrow \mathcal{Z}$ is bounded, then the following statements are equivalent:

• (a) The operator $D^{n}_{\varphi,u} :\mathcal{B}\rightarrow \mathcal{Z}$ is compact.

• (b)
  $$\lim_{|\varphi(w)|\rightarrow1} \bigl\Vert D^{n}_{\varphi,u} f_{\varphi (w)}\bigr\Vert _{\mathcal{Z}}=\lim_{|\varphi(w)|\rightarrow1} \bigl\Vert D^{n}_{\varphi,u} g_{\varphi (w)}\bigr\Vert _{\mathcal{Z}}=\lim_{|\varphi (w)|\rightarrow1} \bigl\Vert D^{n}_{\varphi,u} h_{\varphi (w)}\bigr\Vert _{\mathcal{Z}}=0, $$

  where

  $$\begin{aligned}& f_{\varphi (w)}(z)=\frac{1-|\varphi (w)|^{2}}{1-\overline{\varphi (w)}z}, \qquad g_{\varphi (w)}(z)= \frac{(1-|\varphi (w)|^{2})^{2}}{(1-\overline{\varphi (w)}z)^{2}}, \\& h_{\varphi (w)}(z)=\frac{ (1-|\varphi (w)|^{2})^{3}}{(1-\overline {\varphi (w)}z)^{3}},\quad z\in\mathbb{D}. \end{aligned}$$
• (c)
  $$\begin{aligned} \lim_{|\varphi(z)|\rightarrow 1}\frac{(1-|z|^{2}) |u''(z)| }{ (1-|\varphi(z)|^{2})^{n} } =& \lim_{|\varphi(z)|\rightarrow1} \frac{(1-|z|^{2}) |u(z)||\varphi'(z)|^{2} }{ (1-|\varphi(z)|^{2})^{n+2} } \\ =& \lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2}) |2u'(z)\varphi'(z)+u(z)\varphi''(z)| }{(1-| \varphi(z)|^{2})^{1+n} } =0 . \end{aligned}$$

Motivated by these observations, the purpose of this paper is to give some estimates of the essential norm for the operator $D^{n}_{\varphi ,u}:\mathcal{B}\rightarrow\mathcal{Z}$. Moreover, we give a new characterization for the boundedness, compactness, and essential norm of the operator $D^{n}_{\varphi,u}:\mathcal{B}\rightarrow\mathcal{Z}$.

Throughout this paper, we say that $P\lesssim Q$ if there exists a constant C such that $P\leq CQ$. The symbol $P\approx Q$ means that $P\lesssim Q\lesssim P$.

2 Essential norm of $D^{n}_{\varphi,u}:\mathcal{B} \to \mathcal{Z}$

In this section, we give two estimates of the essential norm for the operator $D^{n}_{\varphi,u}:\mathcal{B} \to\mathcal{Z}$.

Theorem 2.1

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

$$ \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \approx \max \{ A, B, C \}\approx\max \{ E, F, G \} , $$

where

$$\begin{aligned}& A:=\limsup_{|a|\to 1}\biggl\Vert D^{n}_{\varphi,u} \biggl(\frac{1-|a|^{2}}{1-\overline {a}z} \biggr)\biggr\Vert _{\mathcal{Z}}, \qquad B:= \limsup _{|a|\to 1}\biggl\Vert D^{n}_{\varphi,u} \biggl( \frac{(1-|a|^{2})^{2}}{(1-\overline {a}z)^{2}} \biggr)\biggr\Vert _{\mathcal{Z}}, \\& C:= \limsup_{|a|\to 1}\biggl\Vert D^{n}_{\varphi,u} \biggl(\frac{(1-|a|^{2})^{3}}{(1-\overline {a}z)^{3}} \biggr)\biggr\Vert _{\mathcal{Z}}, \qquad F:= \limsup _{|\varphi (z)|\rightarrow1}\frac{(1-|z|^{2}) |u''(z)| }{ (1-|\varphi(z)|^{2})^{n} }, \\& E:=\limsup_{ |\varphi(z)|\rightarrow1}\frac{(1-|z|^{2}) |2u'(z)\varphi'(z)+u(z)\varphi''(z)| }{(1-|\varphi(z)|^{2})^{n+1} }, \end{aligned}$$

and

$$G:= \limsup_{ |\varphi(z)|\rightarrow1}\frac{(1-|z|^{2}) |u(z)||\varphi'(z)|^{2} }{ (1-|\varphi(z)|^{2})^{n+2} }. $$

Proof

First we prove that $\max \{ A, B, C \} \leq\|D^{n}_{\varphi,u}\|_{e,\mathcal {B}\to \mathcal{Z}}$. Let $a\in\mathbb{D}$. Define

$$f_{a}(z)=\frac{1-|a|^{2}}{(1-\overline{a}z)}, \qquad g_{a}(z)= \frac {(1-|a|^{2})^{2}}{(1-\overline{a}z)^{2}}, \qquad h_{a}(z)=\frac{ (1-|a|^{2})^{3}}{(1-\overline{a}z)^{3}}, \quad z\in \mathbb{D}. $$

It is easy to check that $f_{a}, g_{a}, h_{a}\in\mathcal{B}_{0}$ and $\|f_{a}\| _{\mathcal{B}} \lesssim1$, $\|g_{a}\|_{\mathcal{B}} \lesssim1$, $\|h_{a}\|_{\mathcal{B}} \lesssim1$ for all $a\in\mathbb{D}$ and $f_{a}$, $g_{a}$, $h_{a}$ converge to 0 weakly in $\mathcal{B}$ as $|a|\to1$. This follows since a bounded sequence contained in $\mathcal{B}_{0}$ which converges uniformly to 0 on compact subsets of $\mathbb{D}$ converges weakly to 0 in $\mathcal{B}$ (see [37, 42]). Thus, for any compact operator $K: \mathcal{B}_{0}\to\mathcal{Z}$, we have

$$\lim_{|a|\to1}\|Kf_{a}\|_{\mathcal{Z}}=0, \qquad \lim _{|a|\to1}\|Kg_{a}\| _{\mathcal{Z}}=0, \qquad \lim _{|a|\to1}\|Kh_{a}\|_{\mathcal{Z}}=0. $$

Hence

$$\begin{aligned}& \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim \limsup_{|a|\to 1}\bigl\Vert \bigl(D^{n}_{\varphi,u}-K \bigr)f_{a}\bigr\Vert _{\mathcal{Z}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}}} \geq \limsup_{|a|\to1}\bigl\Vert D^{n}_{\varphi,u} f_{a} \bigr\Vert _{\mathcal {Z}}- \limsup_{|a|\to1}\Vert Kf_{a}\Vert _{\mathcal{Z}}= A, \\& \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim \limsup_{|a|\to 1}\bigl\Vert \bigl(D^{n}_{\varphi,u}-K \bigr)g_{a}\bigr\Vert _{\mathcal{Z}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}}} \geq \limsup_{|a|\to1}\bigl\Vert D^{n}_{\varphi,u} g_{a} \bigr\Vert _{\mathcal {Z}}- \limsup_{|a|\to1}\Vert Kg_{a}\Vert _{\mathcal{Z}}=B, \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim&\limsup_{|a|\to 1}\bigl\Vert \bigl(D^{n}_{\varphi,u}-K \bigr)h_{a}\bigr\Vert _{\mathcal{Z}} \\ \geq& \limsup_{|a|\to1}\bigl\Vert D^{n}_{\varphi,u} h_{a} \bigr\Vert _{\mathcal {Z}}-\limsup_{|a|\to1} \Vert Kh_{a}\Vert _{\mathcal{Z}}=C. \end{aligned}$$

Therefore, from the definition of the essential norm, we obtain

$$ \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}}= \inf _{K} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{ \mathcal{B}\to\mathcal{Z}} \gtrsim\max \{ A, B, C \} . $$

Next, we prove that $\|D^{n}_{\varphi,u}\|_{e,\mathcal{B}\to\mathcal{Z}} \gtrsim\max \{ E, F , G \}$. Let $\{z_{j}\}_{j\in\mathbb{N}}$ be a sequence in $\mathbb{D}$ such that $|\varphi(z_{j})|\rightarrow1$ as $j\rightarrow\infty$. Define

$$\begin{aligned}& k_{j}(z) = \frac{1-|\varphi(z_{j})|^{2}}{1-\overline{\varphi (z_{j})}z}-\frac{2n+5}{(n+1)(n+3)} \frac{(1-|\varphi (z_{j})|^{2})^{2}}{(1-\overline{\varphi(z_{j})}z)^{2}} +\frac{2}{(n+1)(n+3)} \frac{(1-|\varphi(z_{j})|^{2})^{3}}{(1-\overline {\varphi(z_{j})}z)^{3}}, \\& l_{j}(z) = \frac{1-|\varphi(z_{j})|^{2}}{1-\overline{\varphi (z_{j})}z}-\frac{2(n+3)}{2+(n+1)(n+4)} \frac{(1-|\varphi (z_{j})|^{2})^{2}}{(1-\overline{\varphi(z_{j})}z)^{2}} +\frac{2}{2+(n+1)(n+4)} \frac{(1-|\varphi(z_{j})|^{2})^{3}}{(1-\overline {\varphi(z_{j})}z)^{3}}, \end{aligned}$$

and

$$ m_{j}(z) = \frac{1-|\varphi(z_{j})|^{2}}{1-\overline{\varphi (z_{j})}z}-\frac{2}{n+1} \frac{(1-|\varphi(z_{j})|^{2})^{2}}{(1-\overline {\varphi(z_{j})}z)^{2}} +\frac{2}{(n+1)(n+2)} \frac{(1-|\varphi(z_{j})|^{2})^{3}}{(1-\overline {\varphi(z_{j})}z)^{3}}. $$

Similarly to the above we see that all $k_{j}$, $l_{j}$, and $m_{j}$ belong to $\mathcal{B}_{0}$ and converge to 0 weakly in $\mathcal{B}$. Moreover,

$$\begin{aligned}& k^{(n)}_{j}\bigl(\varphi(z_{j})\bigr)=0, \qquad k^{(n+2)}_{j}\bigl(\varphi(z_{j})\bigr)=0,\qquad \bigl\vert k^{(n+1)}_{j}\bigl(\varphi(z_{j})\bigr)\bigr\vert =\frac{n!}{n+3}\frac{\vert \varphi (z_{j})\vert ^{n+1}}{(1-\vert \varphi(z_{j})\vert ^{2})^{n+1}}, \\& l^{(n+1)}_{j}\bigl(\varphi(z_{j})\bigr)=0, \qquad l^{(n+2)}_{j}\bigl(\varphi(z_{j})\bigr)=0,\qquad \bigl\vert l^{(n)}_{j}\bigl(\varphi(z_{j})\bigr) \bigr\vert = \frac{2 n!}{2+(n+1)(n+4)}\frac{\vert \varphi (z_{j})\vert ^{n}}{(1-\vert \varphi(z_{j})\vert ^{2})^{n}}, \\& m^{(n)}_{j}\bigl(\varphi(z_{j})\bigr)=0, \qquad m^{(n+1)}_{j}\bigl(\varphi(z_{j})\bigr)=0,\qquad \bigl\vert m^{(n+2)}_{j}\bigl(\varphi(z_{j})\bigr)\bigr\vert =2n!\frac{\vert \varphi (z_{j})\vert ^{n+2}}{(1-\vert \varphi(z_{j})\vert ^{2})^{n+2}}. \end{aligned}$$

Then for any compact operator $K: \mathcal{B} \to \mathcal{Z}$, we obtain

$$\begin{aligned}& \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim \limsup_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(k_{j}) \bigr\Vert _{\mathcal{Z}}- \limsup_{j\rightarrow\infty} \bigl\Vert K(k_{j}) \bigr\Vert _{\mathcal{Z}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}}} \gtrsim \limsup_{j\rightarrow\infty} \frac{(1-|z_{j}|^{2}) |2u'(z_{j})\varphi'(z_{j})+u(z_{j})\varphi''(z_{j})||\varphi (z_{j})|^{n+1}}{(1-|\varphi(z_{j})|^{2})^{n+1}}, \\& \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim \limsup_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(l_{j}) \bigr\Vert _{\mathcal{Z}} - \limsup_{j\rightarrow\infty} \bigl\Vert K(l_{j}) \bigr\Vert _{\mathcal{Z}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}}} \gtrsim \limsup_{j\rightarrow\infty} \frac {(1-|z_{j}|^{2})|u''(z_{j})||\varphi(z_{j})|^{n}}{(1-|\varphi(z_{j})|^{2})^{n}}, \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim& \limsup_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(m_{j}) \bigr\Vert _{\mathcal{Z}} - \limsup_{j\rightarrow\infty} \bigl\Vert K(m_{j}) \bigr\Vert _{\mathcal{Z}} \\ \gtrsim&\limsup_{j\rightarrow\infty} \frac {(1-|z_{j}|^{2})|u(z_{j})||\varphi'(z_{j})|^{2}|\varphi (z_{j})|^{n+2}}{(1-|\varphi(z_{j})|^{2})^{n+2}}. \end{aligned}$$

From the definition of the essential norm, we obtain

$$\begin{aligned}& \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal {Z}} = \inf _{K} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z} } \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal {Z}}}\gtrsim \limsup_{j\rightarrow\infty} \frac{(1-|z_{j}|^{2}) |2u'(z_{j})\varphi'(z_{j})+u(z_{j})\varphi''(z_{j})||\varphi (z_{j})|^{n+1}}{(1-|\varphi(z_{j})|^{2})^{n+1}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal {Z}}}= \limsup_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2}) |2u'(z)\varphi '(z)+u(z)\varphi''(z)|}{(1-|\varphi(z)|^{2})^{n+1}} = E , \\& \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} = \inf _{K} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z} } \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}}}\gtrsim\limsup_{j\rightarrow\infty} \frac{(1-|z_{j}|^{2})|u''(z_{j})||\varphi (z_{j})|^{n}}{(1-|\varphi(z_{j})|^{2})^{n}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}}} = \limsup_{ |\varphi(z)|\rightarrow1}\frac {(1-|z|^{2})|u''(z)|}{(1-|\varphi(z)|^{2})^{n}}=F, \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} = &\inf _{K} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z} } \\ \gtrsim& \limsup_{j\rightarrow\infty} \frac {(1-|z_{j}|^{2})|u(z_{j})||\varphi'(z_{j})|^{2}|\varphi (z_{j})|^{n+2}}{(1-|\varphi(z_{j})|^{2})^{n+2}} \\ =& \limsup_{ |\varphi(z)|\rightarrow1}\frac {(1-|z|^{2})|u(z)||\varphi'(z)|^{2}}{(1-|\varphi (z)|^{2})^{n+2}}=G. \end{aligned}$$

Hence

$$\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \gtrsim \max \{ E, F , G \}. $$

Now, we prove that

$$\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \lesssim \max \{ A, B, C \} \quad \mbox{and} \quad \bigl\Vert D^{n}_{\varphi,u} \bigr\Vert _{e,\mathcal {B}\to \mathcal{Z}} \lesssim\max \{ E, F, G \}. $$

For $r\in[0,1)$, set $K_{r}: H(\mathbb{D})\to H(\mathbb{D})$ by $(K_{r} f)(z)=f_{r}(z)=f(rz)$, $f\in H(\mathbb{D})$. It is obvious that $f_{r}\to f$ uniformly on compact subsets of $\mathbb{D}$ as $r\to1$. Moreover, the operator $K_{r}$ is compact on $\mathcal{B}$ and $\|K_{r}\|_{\mathcal{B}\to\mathcal{B}}\leq1$ (see [37]). Let $\{ r_{j}\}\subset(0,1)$ be a sequence such that $r_{j}\to1$ as $j\to\infty$. Then for all positive integer j, the operator $D^{n}_{\varphi,u} K_{r_{j}}: \mathcal{B}\rightarrow \mathcal {Z}$ is compact. By the definition of the essential norm, we get

$$ \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \leq \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to \mathcal{Z} }. $$

(2.1)

Therefore, we only need to prove that

$$ \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to\mathcal{Z}} \lesssim\max \{ A, B, C \} $$

and

$$ \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to\mathcal{Z}} \lesssim\max \{ E, F, G \} . $$

For any $f\in\mathcal{B}$ such that $\|f\|_{\mathcal{B}}\leq1$, we consider

$$\begin{aligned}& \bigl\Vert \bigl( D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr)f\bigr\Vert _{\mathcal {Z}} \\& \quad = \bigl\vert u(0)f^{(n)}\bigl(\varphi(0)\bigr)-r_{j}^{n}u(0)f^{(n)} \bigl(r_{j}\varphi(0)\bigr)\bigr\vert \\& \qquad {}+\bigl\vert u'(0) (f-f_{r_{j}})^{(n)} \bigl(\varphi(0)\bigr)+u(0) (f-f_{r_{j}})^{(n+1)}\bigl(\varphi (0) \bigr)\varphi'(0)\bigr\vert \\& \qquad {}+\bigl\Vert u\cdot(f-f_{r_{j}})^{(n)}\circ\varphi\bigr\Vert _{*}, \end{aligned}$$

(2.2)

where $\|f\|_{*}=\sup_{z \in\mathbb{D}}(1-|z|^{2}) |f''(z)|$.

It is obvious that

$$ \lim_{j\to\infty}\bigl\vert u(0)f^{(n)}\bigl(\varphi (0) \bigr)-r_{j}^{n}u(0)f^{(n)}\bigl(r_{j} \varphi(0)\bigr)\bigr\vert =0 $$

(2.3)

and

$$ \lim_{j\to\infty}\bigl\vert u'(0) (f-f_{r_{j}})^{(n)} \bigl(\varphi (0)\bigr)+u(0) (f-f_{r_{j}})^{(n+1)}\bigl(\varphi(0) \bigr)\varphi'(0)\bigr\vert =0. $$

(2.4)

Now, we consider

$$\begin{aligned}& \limsup_{j\to\infty}\bigl\Vert u\cdot(f-f_{r_{j}})^{(n)} \circ\varphi\bigr\Vert _{*} \\& \quad \le \limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+1)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi ''(z)\bigr\vert \\& \qquad {}+ \limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+1)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi ''(z)\bigr\vert \\& \qquad {}+\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z) \bigr\vert \\& \qquad {}+ \limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z) \bigr\vert \\& \qquad {}+\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+2)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert \\& \qquad {}+ \limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+2)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert \\& \quad = Q_{1}+Q_{2}+Q_{3}+Q_{4}+Q_{5}+Q_{6}, \end{aligned}$$

(2.5)

where $N\in\mathbb{N }$ is large enough such that $r_{j}\geq\frac {1}{2}$ for all $j\geq N$,

$$\begin{aligned}& Q_{1}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+1)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi ''(z)\bigr\vert , \\& Q_{2}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+1)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi''(z)\bigr\vert , \\& Q_{3}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z) \bigr\vert , \\& Q_{4}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z) \bigr\vert , \\& Q_{5}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+2)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert , \end{aligned}$$

and

$$Q_{6}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+2)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert . $$

Since $D^{n}_{\varphi,u}:\mathcal{B} \to\mathcal{Z}$ is bounded, by Theorem 1 of [12], we see that $u\in\mathcal{Z}$,

$$\widetilde{K}_{1}:=\sup_{z\in\mathbb{D}}\bigl(1-\vert z \vert ^{2}\bigr) \bigl\vert 2u'(z)\varphi '(z)+u(z)\varphi''(z)\bigr\vert < \infty $$

and

$$\widetilde{K}_{2}:=\sup_{z\in\mathbb{D}}\bigl(1-\vert z \vert ^{2}\bigr) \bigl\vert \varphi '(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert < \infty. $$

Since $r^{n+1}_{j}f^{(n+1)}_{r_{j}}\to f^{(n+1)}$, as well as $r^{n+2}_{j}f^{(n+2)}_{r_{j}}\to f^{(n+2)}$ uniformly on compact subsets of $\mathbb{D}$ as $j\to\infty$, we have

$$ Q_{1} \leq\widetilde{K}_{1} \limsup_{j\to\infty} \sup_{\vert w\vert \leq r_{N}} \bigl\vert f^{(n+1)}(w)- r^{n+1}_{j}f^{(n+1)} (r_{j} w)\bigr\vert =0 $$

(2.6)

and

$$ Q_{5} \leq\widetilde{K}_{2} \limsup_{j\to\infty} \sup_{\vert w\vert \leq r_{N}} \bigl\vert f^{(n+2)}(w)- r^{n+2}_{j}f^{(n+2)} (r_{j} w)\bigr\vert =0. $$

(2.7)

Similarly, from the fact that $u \in\mathcal{Z}$ we have

$$ Q_{3} \leq\|u\|_{\mathcal{Z}}\limsup_{j\to\infty}\sup _{\vert w\vert \leq r_{N}} \bigl\vert f^{(n)}(w)- r^{n}_{j}f^{(n)} (r_{j} w)\bigr\vert =0. $$

(2.8)

Next we consider $Q_{2}$. We have $Q_{2}\leq\limsup_{j\to\infty }(S_{1}+S_{2})$, where

$$ S_{1}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n+1)}\bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z)\varphi'(z)+u(z) \varphi''(z)\bigr\vert $$

and

$$ S_{2}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) r^{n+1}_{j}\bigl\vert f^{(n+1)} \bigl(r_{j}\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi''(z)\bigr\vert . $$

First we estimate $S_{1}$. Using the fact that $\|f\|_{\mathcal{B}}\leq 1$ and Theorem 5.4 in [31], we have

$$\begin{aligned} S_{1} =&\sup_{\vert \varphi(z) \vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n+1)}\bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z)\varphi'(z)+u(z) \varphi''(z)\bigr\vert \\ &{} \times\frac{(1-\vert \varphi(z)\vert ^{2})^{n+1}(n+3)}{ \vert \varphi (z)\vert ^{n+1}n!} \frac{ \vert \varphi(z)\vert ^{n+1} n! }{(n+3)(1-\vert \varphi (z)\vert ^{2})^{n+1}} \\ \lesssim&\frac{(n+3) \|f\|_{\mathcal{B}} }{n! r_{N}^{n+1}}\sup_{\vert \varphi(z) \vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert 2u'(z)\varphi'(z)+u(z) \varphi''(z)\bigr\vert \\ &{} \times\frac{n! \vert \varphi(z)\vert ^{n+1} }{(n+3)(1-\vert \varphi (z)\vert ^{2})^{n+1}} \\ \lesssim& \sup_{\vert \varphi(z) \vert >r_{N}} \bigl(1-\vert z\vert ^{2} \bigr) \bigl\vert 2u'(z)\varphi '(z)+u(z) \varphi''(z)\bigr\vert \frac{n! \vert \varphi(z)\vert ^{n+1} }{(n+3)(1-\vert \varphi (z)\vert ^{2})^{n+1}} \\ \lesssim& \sup_{\vert a \vert >r_{N}} \biggl\Vert D^{n}_{\varphi,u} \biggl(f_{a}-\frac {(2n+5) g_{a}}{(n+1)(n+3)}+\frac{2h_{a}}{(n+1)(n+3)} \biggr)\biggr\Vert _{\mathcal{Z}} \\ \lesssim& \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal {Z}}+ \frac{2n+5}{(n+1)(n+3)} \sup _{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} \\ &{} +\frac{2}{(n+1)(n+3)} \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}}. \end{aligned}$$

(2.9)

Taking the limit as $N\to\infty$ we obtain

$$\begin{aligned} \begin{aligned} \limsup_{j\to\infty}S_{1} \lesssim{}& \limsup _{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal{Z}}+ \limsup_{|a|\to 1} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal{Z}} \\ &{} + \limsup_{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal{Z}} \\ = {}& A+B+C. \end{aligned} \end{aligned}$$

Similarly, we have $\limsup_{j\to\infty}S_{2}\lesssim A+B+C$, i.e., we get

$$ Q_{2}\lesssim A+ B+ C \lesssim\max \{ A, B , C \} . $$

(2.10)

From (2.9), we see that

$$\limsup_{j\to\infty}S_{1} \lesssim\limsup _{|\varphi(z)|\rightarrow 1}\frac{(1-|z|^{2}) |2u'(z)\varphi'(z)+u(z)\varphi''(z)|}{(1-|\varphi (z)|^{2})^{n+1}}=E. $$

Similarly we have $\limsup_{j\to\infty}S_{2} \lesssim E$. Therefore

$$ Q_{2} \lesssim E . $$

(2.11)

Next we consider $Q_{4}$. We have $Q_{4}\leq\limsup_{j\to\infty }(S_{3}+S_{4})$, where

$$ S_{3}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z)\bigr\vert $$

and

$$ S_{4}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) r^{n}_{j}\bigl\vert f^{(n)} \bigl(r_{j}\varphi (z)\bigr)\bigr\vert \bigl\vert u''(z)\bigr\vert . $$

After some calculation, we have

$$\begin{aligned} S_{3} =& \sup_{\vert \varphi(z) \vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z)\bigr\vert \\ &{} \times\frac{(1-\vert \varphi(z)\vert ^{2})^{n}(2+(n+1)(n+4))}{2n! \vert \varphi(z)\vert ^{n}} \frac{2}{2+(n+1)(n+4)} \frac{n! \vert \varphi(z)\vert ^{n} }{(1-\vert \varphi (z)\vert ^{2})^{n}} \\ \lesssim&\frac{ 2^{n}(2+(n+1)(n+4))}{2n! } \|f\|_{\mathcal{B}} \sup_{\vert \varphi (z) \vert >r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert u''(z) \bigr\vert \\ &{} \times \frac{2}{2+(n+1)(n+4)} \frac{n!\vert \varphi(z)\vert ^{n} }{(1-\vert \varphi (z)\vert ^{2})^{n}} \\ \lesssim& \sup_{\vert \varphi(z) \vert >r_{N}} \frac{2n! }{2+(n+1)(n+4)} \frac {(1-\vert z\vert ^{2}) \vert u''(z)\vert \vert \varphi(z)\vert ^{n} }{(1-\vert \varphi(z)\vert ^{2})^{n}} \\ \lesssim& \sup_{\vert a\vert >r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal{Z}}+ \frac{2(n+3)}{2+(n+1)(n+4)} \sup _{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} \\ &{} + \frac{2}{2+(n+1)(n+4)} \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}} \\ \lesssim& \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal {Z}}+ \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}}+ \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}}. \end{aligned}$$

(2.12)

Taking the limit as $N\to\infty$ we obtain

$$\begin{aligned} \limsup_{j\to\infty}S_{3} \lesssim& \limsup _{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal{Z}}+ \limsup_{|a|\to 1} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} \\ &{} + \limsup_{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}} \\ =& A+B+C. \end{aligned}$$

Similarly, we have $\limsup_{j\to\infty}S_{4} \lesssim A+B+C$, i.e., we get

$$ Q_{4} \lesssim A + B + C\lesssim\max \{ A, B , C \} . $$

(2.13)

From (2.12), we see that

$$\limsup_{j\to\infty}S_{3} \lesssim\limsup _{ |\varphi(z)|\rightarrow 1}\frac{(1-|z|^{2})^{\beta}|u''(z)|}{(1-|\varphi(z)|^{2})^{n}}=F. $$

Similarly we have $\limsup_{j\to\infty}S_{4} \lesssim F$. Therefore

$$ Q_{4} \lesssim F . $$

(2.14)

Finally we consider $Q_{6}$. We have $Q_{6}\leq\limsup_{j\to\infty }(S_{5}+S_{6})$, where

$$ S_{5}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n+2)}\bigl(\varphi (z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert $$

and

$$ S_{6}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) r^{n+2}_{j}\bigl\vert f^{(n+2)} \bigl(r_{j}\varphi (z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert . $$

After some calculation, we have

$$\begin{aligned} S_{5} \lesssim& \frac{ 2^{n+2} \|f\|_{\mathcal{B}} }{2n! } \sup _{\vert \varphi(z) \vert >r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert \frac{ 2n!\vert \varphi(z)\vert ^{n+2} }{(1-\vert \varphi(z)\vert ^{2})^{n+2}} \\ \lesssim& \frac{ 2^{n+2}}{2n! } \sup_{\vert \varphi(z) \vert >r_{N}} \bigl(1-\vert z \vert ^{2}\bigr) \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert \frac{ 2n!\vert \varphi(z)\vert ^{n+2} }{(1-\vert \varphi (z)\vert ^{2})^{n+2}} \\ \lesssim& \sup_{\vert a \vert >r_{N}} \biggl\Vert D^{n}_{\varphi,u} \biggl(f_{a}- \frac{2}{n+1} g_{a}+ \frac {2}{(n+1)(n+2)} h_{a} \biggr)\biggr\Vert _{\mathcal{Z}} \\ \lesssim& \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal {Z}}+ \frac{2}{n+1} \sup _{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} + \frac{2}{(n+1)(n+2)} \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}} \\ \leq& \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal {Z}}+ \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}}+ \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}}. \end{aligned}$$

(2.15)

Taking the limit as $N\to\infty$ we obtain

$$\begin{aligned} \begin{aligned} \limsup_{j\to\infty}S_{5} \lesssim{}& \limsup _{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal{Z}}+ \limsup_{|a|\to 1} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} \\ &{} + \limsup_{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}} \\ ={}& A+B+C. \end{aligned} \end{aligned}$$

Similarly, we have $\limsup_{j\to\infty}S_{6} \lesssim A+B+C$, i.e., we get

$$ Q_{6} \lesssim A + B + C\lesssim\max \{ A, B , C \} . $$

(2.16)

From (2.15), we see that

$$\limsup_{j\to\infty}S_{5} \lesssim\limsup _{ |\varphi(z)|\rightarrow 1}\frac{(1-|z|^{2})|\varphi'(z)|^{2}|u(z)|}{(1-|\varphi(z)|^{2})^{n+2}}=G. $$

Similarly we have $\limsup_{j\to\infty}S_{6} \lesssim G$. Therefore

$$ Q_{6} \lesssim G . $$

(2.17)

Hence, by (2.2), (2.3), (2.4), (2.5), (2.6), (2.7), (2.8), (2.10), (2.13), and (2.16) we get

$$\begin{aligned}& \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to\mathcal{Z} } \\& \quad = \limsup_{j\to\infty}\sup_{ \Vert f\Vert _{\mathcal{B}} \leq1}\bigl\Vert \bigl(D^{n}_{\varphi ,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr)f\bigr\Vert _{\mathcal{Z} } \\& \quad = \limsup_{j\to\infty}\sup_{ \Vert f\Vert _{\mathcal{B}} \leq1} \bigl\Vert u\cdot (f-f_{r_{j}})^{(n)}\circ\varphi\bigr\Vert _{*} \lesssim\max \{ A, B, C \}. \end{aligned}$$

(2.18)

Similarly, by (2.2), (2.3), (2.4), (2.5), (2.6), (2.7), (2.8), (2.11), (2.14), and (2.17) we get

$$ \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}-D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to\mathcal{Z}}\lesssim\max \{ E, F, G \} . $$

(2.19)

Therefore, by (2.1), (2.18), and (2.19), we obtain

$$ \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \lesssim \max \{ A, B, C \} \quad \mbox{and} \quad \bigl\Vert D^{n}_{\varphi,u} \bigr\Vert _{e,\mathcal {B}\to \mathcal{Z}} \lesssim\max \{ E, F, G \} . $$

This completes the proof of Theorem 2.1. □

3 New characterization of $D^{n}_{\varphi,u}: \mathcal{B}\to \mathcal{Z}$

In this section, we give a new characterization for the boundedness, compactness, and essential norm of the operator $D^{n}_{\varphi ,u}:\mathcal{B} \to\mathcal{Z}$. For this purpose, we present some definitions and some lemmas which will be used later.

The weighted space, denoted by $H^{\infty}_{v}$, consists of all $f\in H(\mathbb{D})$ such that

$$\|f\|_{v}=\sup_{z \in\mathbb{D}}v(z)\bigl\vert f(z)\bigr\vert < \infty, $$

where $v:\mathbb{D}\rightarrow R_{+}$ is a continuous, strictly positive, and bounded function. $H^{\infty}_{v}$ is a Banach space under the norm $\| \cdot\|_{v}$. The weighted v is called radial if $v(z)=v(|z|)$ for all $z\in\mathbb{D}$. The associated weight ṽ of v is as follows:

$$\tilde{v}=\bigl(\sup\bigl\{ \bigl\vert f(z)\bigr\vert : f\in H^{\infty}_{v}, \|f\|_{v} \leq1 \bigr\} \bigr)^{-1},\quad z\in\mathbb{D}. $$

When $v=v_{\alpha}(z)=(1-|z|^{2})^{\alpha}$ ($0<\alpha <\infty$), it is well known that $\tilde{v}_{\alpha}(z)=v_{\alpha}(z)$. In this case, we denote $H^{\infty}_{v}$ by $H^{\infty}_{v_{\alpha}}$.

Lemma 3.1

[33]

For $\alpha >0$, we have $\lim_{k\rightarrow\infty}k^{\alpha}\|z^{k-1}\|_{v_{\alpha}}=(\frac {2\alpha }{e})^{\alpha}$.

Lemma 3.2

[51]

Let v and w be radial, non-increasing weights tending to zero at the boundary of $\mathbb {D}$. Then the following statements hold.

• (a) The weighted composition operator $uC_{\varphi}:H_{v}^{\infty }\rightarrow H_{w}^{\infty}$ is bounded if and only if $\sup_{z\in \mathbb{D}}\frac{w(z)}{\tilde{v}(\varphi(z))}|u(z)|<\infty$. Moreover, the following holds:

  $$ \|uC_{\varphi}\|_{H_{v}^{\infty}\rightarrow H_{w}^{\infty}}=\sup_{z\in\mathbb{D}} \frac{w(z)}{\tilde{v}(\varphi (z))}\bigl\vert u(z)\bigr\vert . $$

• (b) Suppose $uC_{\varphi}:H_{v}^{\infty}\rightarrow H_{w}^{\infty}$ is bounded. Then

  $$ \|uC_{\varphi}\|_{e, H_{v}^{\infty}\rightarrow H_{w}^{\infty }}=\lim_{s\to1^{-}}\sup _{|\varphi(z)|>s}\frac{w(z)}{\tilde {v}(\varphi(z))}\bigl\vert u(z)\bigr\vert . $$

Lemma 3.3

[52]

Let v and w be radial, non-increasing weights tending to zero at the boundary of $\mathbb {D}$. Then the following statements hold.

• (a) $uC_{\varphi}:H_{v}^{\infty}\rightarrow H_{w}^{\infty}$ is bounded if and only if $\sup_{k\geq0}\frac{\|u \varphi^{k}\|_{w}}{\|z^{k}\|_{v}}<\infty$, with the norm comparable to the above supremum.

• (b) Suppose $uC_{\varphi}:H_{v}^{\infty}\rightarrow H_{w}^{\infty}$ is bounded. Then

  $$ \|uC_{\varphi}\|_{e,H_{v}^{\infty}\rightarrow H_{w}^{\infty} }=\limsup_{k\to \infty} \frac{\|u \varphi^{k}\|_{w}}{\|z^{k}\|_{v}}. $$

Theorem 3.1

Let n be a positive integer, $u \in H(\mathbb{D})$, and φ be an analytic self-map of $\mathbb{D}$. Then the operator $D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$ is bounded if and only if

$$ \left \{ \textstyle\begin{array}{l} \sup_{j\geq1} j^{n+1}\|(2u'\varphi'+u\varphi'')\varphi^{j-1}\| _{v_{1}}< \infty, \\ \sup_{j\geq1} j^{n}\|u''\varphi^{j-1}\|_{v_{1}}< \infty, \\ \sup_{j\geq1} j^{n+2}\|u\varphi^{\prime 2}\varphi^{j-1}\|_{v_{1}}< \infty. \end{array}\displaystyle \right . $$

(3.1)

Proof

By [16], $D^{n}_{\varphi, u} : \mathcal{B} \to\mathcal {Z}$ is bounded if and only if

$$ \left \{ \textstyle\begin{array}{l} \sup_{z\in\mathbb{D}}\frac{(1-|z|^{2}) |2u'(z)\varphi '(z)+u(z)\varphi''(z)|}{(1-|\varphi(z)|^{2})^{n+1}}< \infty, \\ \sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})|u''(z)|}{(1-|\varphi (z)|^{2})^{n}}< \infty, \\ \sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})|u(z)||\varphi '(z)|^{2}}{(1-|\varphi(z)|^{2})^{n+2}}< \infty. \end{array}\displaystyle \right . $$

(3.2)

By Lemma 3.2, the first inequality in (3.2) is equivalent to the weighted composition operator $(2u'\varphi'+u\varphi'')C_{\varphi}: H^{\infty}_{v_{n+1}}\rightarrow H^{\infty}_{v_{1}}$ is bounded. By Lemma 3.3, this is equivalent to

$$\sup_{j\geq1}\frac{\|(2u'\varphi'+u\varphi'') \varphi^{j-1}\| _{v_{1}}}{\|z^{j-1}\|_{v_{1+n}}}< \infty. $$

The second inequality in (3.2) is equivalent to the operator $u''C_{\varphi}: H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}$ is bounded. By Lemma 3.3, this is equivalent to

$$\sup_{j\geq1}\frac{\|u'' \varphi^{j-1}\|_{v_{1}}}{\|z^{j-1}\| _{v_{n}}}< \infty. $$

The third inequality in (3.2) is equivalent to the operator $u\varphi^{\prime 2}C_{\varphi}: H^{\infty}_{v_{n+2}}\rightarrow H^{\infty}_{v_{1}}$ is bounded. By Lemma 3.3, this is equivalent to

$$\sup_{j\geq1}\frac{\|u\varphi^{\prime 2} \varphi^{j-1}\|_{v_{1}}}{\|z^{j-1}\| _{v_{n+2}}}< \infty. $$

By Lemma 3.1, we see that $D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$ is bounded if and only if

$$\begin{aligned}& \sup_{j\geq1} j^{n+1}\bigl\Vert \bigl(2u' \varphi'+u\varphi''\bigr) \varphi^{j-1}\bigr\Vert _{v_{1}} \approx\sup _{j\geq1}\frac{j^{n+1}\Vert (2u'\varphi'+u\varphi '') \varphi^{j-1}\Vert _{v_{1}}}{j^{n+1}\Vert z^{j-1}\Vert _{v_{1+n}}}< \infty, \\& \sup_{j\geq1} j^{n}\bigl\Vert u'' \varphi^{j-1}\bigr\Vert _{v_{1}} \approx\sup _{j\geq1}\frac{j^{n}\Vert u'' \varphi^{j-1}\Vert _{v_{1}}}{j^{n}\Vert z^{j-1}\Vert _{v_{n}}} < \infty, \end{aligned}$$

and

$$\sup_{j\geq1} j^{ n+2}\bigl\Vert u \varphi^{\prime 2} \varphi^{j-1}\bigr\Vert _{v_{1}} \approx \sup_{j\geq1}\frac{j^{ n+2}\Vert u\varphi^{\prime 2} \varphi^{j-1}\Vert _{v_{1}}}{j^{ n+2}\Vert z^{j-1}\Vert _{v_{ n+2}}}< \infty. $$

The proof is completed. □

Theorem 3.2

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

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \approx \max \{M_{1}, M_{2}, M_{3} \}, $$

where

$$\begin{aligned}& M_{1}:=\limsup_{j\rightarrow\infty} j^{1+n}\bigl\Vert \bigl(2u'\varphi'+u\varphi '' \bigr)\varphi^{j-1}\bigr\Vert _{v_{1}}, \\& M_{2}:= \limsup_{j\rightarrow\infty} j^{n}\bigl\Vert u''\varphi^{j-1}\bigr\Vert _{v_{1}},\qquad M_{3}:=\limsup_{j\rightarrow\infty} j^{ n+2}\bigl\Vert u \bigl(\varphi'\bigr)^{2}\varphi ^{j-1}\bigr\Vert _{v_{1}}. \end{aligned}$$

Proof

From the proof of Theorem 3.1 we know that the boundedness of $D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$ is equivalent to the boundedness of the operators $(2u'\varphi'+u\varphi'')C_{\varphi}: H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}$, $u''C_{\varphi}: H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}$, and $u\varphi^{\prime 2}C_{\varphi}: H^{\infty}_{v_{ n+2}}\rightarrow H^{\infty}_{v_{1}}$.

The upper estimate. By Lemmas 3.1 and 3.3, we get

$$\begin{aligned}& \bigl\Vert \bigl(2u'\varphi'+u\varphi'' \bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}} = \limsup _{j\rightarrow \infty}\frac{\Vert (2u'\varphi'+u\varphi'') \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{1+n}}} \\& \hphantom{\bigl\Vert \bigl(2u'\varphi'+u\varphi'' \bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}}}= \limsup_{j\rightarrow\infty}\frac{j^{1+n}\Vert (2u'\varphi '+u\varphi'') \varphi^{j-1}\Vert _{v_{1}}}{j^{1+n}\Vert z^{j-1}\Vert _{v_{1+n}}} \\& \hphantom{\bigl\Vert \bigl(2u'\varphi'+u\varphi'' \bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}}}\approx \limsup_{j\rightarrow\infty}j^{1+n}\bigl\Vert \bigl(2u'\varphi'+u\varphi'' \bigr) \varphi^{j-1}\bigr\Vert _{v_{1}}, \\& \bigl\Vert u''C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}} = \limsup_{j\rightarrow\infty}\frac{\Vert u'' \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{n}}}=\limsup _{j\rightarrow\infty}\frac{j^{\alpha +n-1}\Vert u'' \varphi^{j-1}\Vert _{v_{1}}}{j^{\alpha +n-1}\Vert z^{j-1}\Vert _{v_{n}}} \\& \hphantom{\bigl\Vert u''C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}}}\approx \limsup_{j\rightarrow\infty}j^{n}\bigl\Vert u'' \varphi^{j-1}\bigr\Vert _{v_{1}}, \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert u\varphi^{\prime 2}C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{ n+2}}\rightarrow H^{\infty}_{v_{1}}} =& \limsup_{j\rightarrow\infty}\frac{\Vert u\varphi ^{\prime 2} \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{ n+2}}}=\limsup _{j\rightarrow\infty}\frac{j^{ n+2}\Vert u\varphi^{\prime 2} \varphi^{j-1}\Vert _{v_{1}}}{j^{ n+2}\Vert z^{j-1}\Vert _{v_{ n+2}}} \\ \approx& \limsup_{j\rightarrow\infty}j^{ n+2}\bigl\Vert u \varphi^{\prime 2} \varphi^{j-1}\bigr\Vert _{v_{1}}. \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \lesssim{}& \bigl\Vert \bigl(2u'\varphi'+u \varphi''\bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}}+ \bigl\Vert u''C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}} \\ &{}+ \bigl\Vert u\varphi^{\prime 2}C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{ n+2}}\rightarrow H^{\infty}_{v_{1}}} \\ \lesssim{}& \max \{M_{1}, M_{2}, M_{3} \}. \end{aligned} \end{aligned}$$

The lower estimate. From Theorem 2.1, and Lemmas 3.1 and 3.2, we have

$$\begin{aligned}& \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \gtrsim E = \bigl\Vert \bigl(2u'\varphi'+u \varphi''\bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} }} = \limsup_{j\rightarrow\infty}\frac{\Vert (2u'\varphi'+u\varphi '')\varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{1+n}}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} }} \approx \limsup_{j\rightarrow\infty}j^{\alpha +n}\bigl\Vert \bigl(2u'\varphi '+u\varphi'' \bigr) \varphi^{j-1}\bigr\Vert _{v_{1}} , \\& \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \gtrsim F =\bigl\Vert u''C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}}=\limsup_{j\rightarrow\infty }\frac{\Vert u'' \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{n}}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} }} \approx \limsup_{j\rightarrow\infty}j^{\alpha +n-1}\bigl\Vert u'' \varphi^{j-1}\bigr\Vert _{v_{1}}, \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \gtrsim&G = \bigl\Vert u\varphi^{\prime 2}C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{ n+2}}\rightarrow H^{\infty}_{v_{1}}}= \limsup_{j\rightarrow\infty }\frac{\Vert u\varphi^{\prime 2} \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{ n+2}}} \\ \approx& \limsup_{j\rightarrow\infty}j^{ n+2}\bigl\Vert u \varphi^{\prime 2} \varphi^{j-1}\bigr\Vert _{v_{1}}. \end{aligned}$$

Therefore $\| D^{n}_{\varphi, u}\|_{e, \mathcal{B} \to \mathcal{Z} } \gtrsim\max \{M_{1}, M_{2}, M_{3} \}$. This completes the proof. □

From Theorem 3.2, we immediately get the following result.

Theorem 3.3

Let n be a positive integer, $u \in H(\mathbb{D})$, and φ be an analytic self-map of $\mathbb{D}$ such that $D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$ is bounded. Then $D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$ is compact if and only if

$$\begin{aligned}& \limsup_{j\rightarrow\infty} j^{1+n}\bigl\Vert \bigl(2u'\varphi'+u\varphi '' \bigr)\varphi^{j-1}\bigr\Vert _{v_{1}}=0, \qquad \limsup_{j\rightarrow\infty} j^{n}\bigl\Vert u''\varphi^{j-1}\bigr\Vert _{v_{1}}=0, \end{aligned}$$

and

$$\limsup_{j\rightarrow\infty} j^{ n+2}\bigl\Vert u\bigl( \varphi'\bigr)^{2}\varphi^{j-1}\bigr\Vert _{v_{1}}=0. $$

4 Conclusion

The boundedness and compactness of $D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$ were characterized in [12] and [16]. In this paper, we give a new characterization for the boundedness and compactness of $D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$. Moreover, using the method in [21], we completely characterize the essential norm of $D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$.