# 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.

## 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 [19]. 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, [1221]. Recently there has been a huge interest in the study of various related product-type operators containing composition operators; see, e.g., [2230] 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, 3245].

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, 4650] 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:

1. (a)

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

2. (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}
3. (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 $$P\lesssim Q\lesssim P$$.

## 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. □

## 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 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.

1. (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:

$$\|uC_{\varphi}\|_{H_{v}^{\infty}\rightarrow H_{w}^{\infty}}=\sup_{z\in\mathbb{D}} \frac{w(z)}{\tilde{v}(\varphi (z))}\bigl\vert u(z)\bigr\vert .$$
2. (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.

1. (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.

2. (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

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

## 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 $$D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z}$$.

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## Acknowledgements

This project is partially supported by the Macao Science and Technology Development Fund (No. 083/2014/A2).

## Author information

Authors

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Correspondence to Xiangling Zhu.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The four authors contributed equally to the writing of this paper. They read and approved the final version of the manuscript.

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Hu, Q., Shi, Y., Shi, Y. et al. Essential norm of generalized weighted composition operators from the Bloch space to the Zygmund space. J Inequal Appl 2016, 123 (2016). https://doi.org/10.1186/s13660-016-1066-4

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• DOI: https://doi.org/10.1186/s13660-016-1066-4

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### Keywords

• Bloch space
• Zygmund space
• essential norm
• generalized weighted composition operator