Essential Norm of Weighted Composition Operators from the Lipschtiz Space to the Zygmund Space

Abstract

In this paper, we give some estimates for the essential norm of weighted composition operators from the Lipschitz space to the Zygmund space.

1 Introduction

Let ${\mathbb {D}}$ be the open unit disk in the complex plane ${\mathbb C}$. Let $H({\mathbb {D}})$ denote the space of analytic functions on ${\mathbb D}$. Let $S({\mathbb D})$ denote the set of all analytic self-maps of ${\mathbb {D}}$.

Let $\varphi \in S({\mathbb {D}})$ and $u \in H({\mathbb {D}})$. The weighted composition operator, denoted by $uC_\varphi$, is defined as follows:

$$\begin{aligned} uC_\varphi f =u(z)\cdot f(\varphi (z)), \ \ f \in H({\mathbb {D}}). \end{aligned}$$

When $u=1$, we get the composition operator $C_\varphi$. When $\varphi (z)=z$, we get the multiplication operator $M_u$. The reader can refer to [6, 33] for more information of the theory of composition operator.

Two well-known classes of small spaces of analytic functions that will be considered in this work are the Lipschitz space $\mathrm{Lip}_\alpha$ and the Zygmund space. For $0<\alpha <1$, the Lipschitz space $\mathrm{Lip}_\alpha$ consists of all functions $f\in H({\mathbb D})$ satisfying the Lipschitz condition of order $\alpha$, i.e., there exists a constant $C>0$ such that

$$\begin{aligned} |f(z)-f(w)|\le C|z-w|^\alpha ,\quad \hbox { for all }\quad z,w\in {\mathbb D}. \end{aligned}$$

${\mathrm{Lip}_\alpha }$ is a Banach space with the following norm

$$\begin{aligned} \Vert f\Vert _{\mathrm{Lip}_\alpha }=|f(0)|+\sup \left\{ \frac{|f(z)-f(w)|}{|z-w|^\alpha }: z,w\in {\mathbb {D}}, z\ne w\right\} <\infty . \end{aligned}$$

Moreover, $\Vert f\Vert _\infty \le \Vert f\Vert _{\mathrm{Lip}_\alpha }.$ By a theorem of Hardy and Littlewood, an $f\in H({\mathbb D})$ belongs to $\mathrm{Lip}_\alpha$ if and only if

$$\begin{aligned} \alpha (f)= \sup _{z \in {\mathbb D}}(1-|z|^2)^{1-\alpha } |f'(z)|<\infty \end{aligned}$$

(see [24] for a complete proof and an extension of the result, as well as [2] for some closely related results). Moreover, $\Vert f\Vert _{\mathrm{Lip}_\alpha } \approx |f(0)|+\alpha (f)$.

The Bloch space, denoted by ${\mathcal {B}}={\mathcal {B}}({\mathbb D})$, is defined as follows:

$$\begin{aligned} {\mathcal {B}}=\Big \{f\in H({\mathbb D}): \Vert f\Vert _{\mathcal {B}} =|f(0)|+ \sup _{z \in {\mathbb D}}\big (1-|z|^2\big ) |f'(z)|<\infty \Big \}. \end{aligned}$$

The Zygmund space, denoted by ${\mathcal {Z}}$, consisting of all $f \in H({\mathbb D})\cap C(\overline{{\mathbb D}})$ such that (see [12])

$$\begin{aligned} \sup \frac{\left| f\big (e^{i(\theta +h)}\big )+f\big (e^{i(\theta -h)}\big )-2f\big (e^{i\theta }\big )\right| }{h}<\infty , \end{aligned}$$

where the supremum is taken over all $\theta \in {\mathbb R}$ and $h>0$. As a consequence of Theorem 5.3 of [7] and the Closed Graph Theorem, an $f\in H({\mathbb D})$ belongs to ${\mathcal {Z}}$ if and only if

$$\begin{aligned} \Vert f\Vert _\beta = \sup _{z\in {\mathbb D}}\big (1-|z|^2\big )|f''(z)|<\infty . \end{aligned}$$

${\mathcal {Z}}$ is a Banach space with the following norm

$$\begin{aligned} \Vert f\Vert _{\mathcal {Z}} = |f(0)|+|f'(0)|+\sup _{z\in {\mathbb D}}\big (1-|z|^2\big )|f''(z)|. \end{aligned}$$

The spaces $\mathrm{Lip}_\alpha$ and the Zygmund space play an important role in connection to the theory of the $H^p$ spaces when $0<p<1$. For more information on these facts, we refer the interested reader to [7]. See [1, 4, 5, 8, 9, 11–17, 28, 29] for some results of composition operators, weighted composition operators, and related operators on the Zygmund space. For some extensions of the Zygmund space on other domains and operators on them, see, [25–29].

Madigan and Matheson in [19] characterized the compactness of $C_\varphi : \mathcal {B} \rightarrow \mathcal {B}$, while Montes-Rodrieguez in [21] studied the essential norm for the operator $C_{\varphi }: \mathcal {B} \rightarrow \mathcal {B}$, i.e., he obtained

$$\begin{aligned} \Vert C_{\varphi }\Vert _{e, \mathcal {B} \rightarrow \mathcal {B} }=\limsup _{|\varphi (z)|\rightarrow 1} \frac{(1-|z |^2) |\varphi '(z)|}{ 1-|\varphi (z)|^2 }. \end{aligned}$$

Recall that 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,

$$\begin{aligned} \Vert T\Vert _{e, X\rightarrow Y}=\inf \{\Vert T-K\Vert _{X\rightarrow Y}: K~\text{ is } \text{ compact }\}. \end{aligned}$$

Here X and Y be Banach spaces, $\Vert \cdot \Vert _{X\rightarrow Y}$ denote the operator norm. Tjani in [30] proved that $C_\varphi : \mathcal {B} \rightarrow \mathcal {B}$ is compact if and only if

$$\begin{aligned} \lim _{|a|\rightarrow 1} \Vert C_\varphi \left( \frac{a-z}{1-\bar{a}z}\right) \Vert _ \mathcal {B} =0 . \end{aligned}$$

Wulan, Zheng, and Zhu in [31] obtained a new characterization of the compactness of $C_\varphi : \mathcal {B} \rightarrow \mathcal {B}$, i.e., they showed that $C_\varphi : \mathcal {B} \rightarrow \mathcal {B}$ is compact if and only if $\lim _{n\rightarrow \infty }\Vert \varphi ^n \Vert _ \mathcal {B} =0.$ Zhao in [32] showed that

$$\begin{aligned} \Vert C_\varphi \Vert _{e, \mathcal {B} \rightarrow \mathcal {B} } = \frac{e}{2 } \limsup _{n\rightarrow \infty } \Vert \varphi ^n \Vert _{ \mathcal {B} }. \end{aligned}$$

In [22], the authors studied the boundedness and compactness of the operator $u C_\varphi :{\mathcal {B}} \rightarrow {\mathcal {B}}$. In [3], Colonna gave a new characterization for the boundedness and compactness of the operator $u C_\varphi :{\mathcal {B}} \rightarrow {\mathcal {B}}$ by using $\Vert u\varphi ^n\Vert _ \mathcal {B}$. The essential norm of the operator $u C_\varphi :{\mathcal {B}} \rightarrow {\mathcal {B}}$ was studied in [10, 18, 20]. For some other results on composition and related operators mapping into the Bloch space see, e.g., [3, 13, 18–20, 22, 23, 28, 29, 32] and the related references therein.

In [5], Colonna and Li studied the weighted composition operator $uC_\varphi : \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$. They show that $uC_\varphi :\mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$ is bounded (respectively, compact) if and only if the sequence $\{ \Vert k^{-\alpha }u\varphi ^k\Vert _{\mathcal {Z}}\}$ is bounded (respectively, $uC_\varphi$ is bounded and $\{ \Vert k^{-\alpha }u\varphi ^k\Vert _{\mathcal {Z}}\}$ converges to 0 as $k\rightarrow \infty$). Among others, they obtained the following result.

Theorem A

Let $0<\alpha <1$, $u \in H({\mathbb D})$, $\varphi \in S({\mathbb D})$ and suppose that $uC_\varphi : \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$ is bounded. Then the following conditions are equivalent:

• (a) The operator $uC_\varphi : \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$ is compact.

• (b) $\displaystyle \lim _{k\rightarrow \infty } \Vert k^{-\alpha }u \varphi ^k\Vert _{\mathcal {Z}}=0.$

• (c) $\displaystyle \lim _{|\varphi (w)|\rightarrow 1} \Vert uC_\varphi f_{\varphi (w),j}\Vert _{\mathcal {Z} } =0, ~~j=1,2,3,$ where

  $$\begin{aligned} f_{a,j}(z)= \frac{(1-|a|^2)^j}{(1-\overline{a}z)^{j-\alpha }} : \ a\in {\mathbb D},\ j=1,2,3. \end{aligned}$$

• (d) $\lim \limits _{|\varphi (z)|\rightarrow 1}\frac{(1-|z|^{2}) |u(z)||\varphi '(z)|^2 }{ (1-|\varphi (z)|^2)^{2-\alpha }}=0,\ \lim \limits _{|\varphi (z)|\rightarrow 1}\frac{(1-|z|^2) |2u'(z)\varphi '(z)+u(z)\varphi ''(z)| }{(1-|\varphi (z)|^2)^{1-\alpha } } =0.$

Motivated by the above work, we give the corresponding estimates for the essential norm of the weighted composition operator $uC_\varphi : \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$.

Throughout this paper, we shall adopt the convention of denoting by C a positive constant whose value may change at each occurrence. We say that $A\lesssim B$ if there exists a constant C such that $A\le CB$. The symbol $A\approx B$ means that $A\lesssim B\lesssim A$.

2 Main Results and Proofs

In this section, we give some estimates for the essential norm of the operator $uC_\varphi : \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$. For this purpose, we need to state some lemmas which will be used in the proof of main results in this paper.

Lemma 2.1

[30] Let X, Y be two Banach spaces of analytic functions on ${\mathbb D}$. Suppose that

• (1) The point evaluation functionals on Y are continuous.

• (2) The closed unit ball of X is a compact subset of X in the topology of uniform convergence on compact sets.

• (3) $T : X\rightarrow Y$ is continuous when X and Y are given the topology of uniform convergence on compact sets.

Then, T is a compact operator if and only if given a bounded sequence $\{f_n\}$ in X such that $f_n\rightarrow 0$ uniformly on compact sets, then the sequence $\{Tf_n\}$ converges to zero in the norm of Y.

Lemma 2.2

[22] Let $0<\alpha <1$ and let $\{f_n\}$ be a bounded sequence in $\mathrm{Lip}_\alpha$ which converges to zero uniformly on compact subsets of ${\mathbb {D}}$. Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{z\in {\mathbb {D}}}|f_n(z)|=0. \end{aligned}$$

Theorem 2.1

Let $0<\alpha <1,$ $u \in H({\mathbb {D}})$ and $\varphi \in S({\mathbb D})$ with $\Vert \varphi \Vert _\infty =1$ such that $uC_\varphi : \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$ is bounded. Then

$$\begin{aligned} \Vert uC_{\varphi }\Vert _{e, \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \approx \max \big \{ A, B , C\big \} \approx \big \{ E, F \big \}, \end{aligned}$$

where

$$\begin{aligned} A:= & {} \limsup _{|a|\rightarrow 1 }\left\| uC_{\varphi }\left( \frac{1-|a|^2}{(1-\bar{a}z)^{1-\alpha }}\right) \right\| _{{\mathcal {Z}} }, \\ B:= & {} \limsup _{|a|\rightarrow 1 }\left\| uC_{\varphi }\left( \frac{(1-|a|^2)^2}{(1-\bar{a}z)^{2-\alpha }} \right) \right\| _{\mathcal {Z}},\\ C:= & {} \limsup _{|a|\rightarrow 1 }\left\| uC_{\varphi }\left( \frac{(1-|a|^2)^3}{(1-\bar{a}z)^{3-\alpha }} \right) \right\| _{{\mathcal {Z}} },\\ E:= & {} \limsup _{|\varphi (z)|\rightarrow 1}\frac{(1-|z|^{2}) |u(z)||\varphi '(z)|^2 }{ (1-|\varphi (z)|^2)^{2-\alpha }}, \end{aligned}$$

and

$$\begin{aligned} F:= & {} \limsup _{|\varphi (z)|\rightarrow 1}\frac{(1-|z|^2) |2u'(z)\varphi '(z)+u(z)\varphi ''(z)| }{(1-|\varphi (z)|^2)^{1-\alpha } }. \end{aligned}$$

Proof

The lower estimate

First we prove that

$$\begin{aligned} \max \big \{ A, B , C\big \} \lesssim \Vert uC_\varphi \Vert _{e, \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } . \end{aligned}$$

Let $a\in {\mathbb D}$. It is easy to check that $\Vert f_{a,j}\Vert _{ \mathrm{Lip}_\alpha }<\infty , j=1,2,3,$ for all $a\in {\mathbb D}$. Moreover, $f_{a,1}, f_{a,2}, f_{a,3}$ converge to zero uniformly on compact subsets of ${\mathbb D}$ as $|a|\rightarrow 1$. Thus, for any compact operator $K: \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$, by Lemma 2.1 we have

$$\begin{aligned} \lim _{|a|\rightarrow 1}\Vert K (f_{a,1})\Vert _{{\mathcal {Z}} }=0, ~~~\lim _{|a|\rightarrow 1}\Vert K(f_{a,2})\Vert _{{\mathcal {Z}} }=0,~~~\lim _{|a|\rightarrow 1}\Vert K(f_{a,3})\Vert _{{\mathcal {Z}} }=0. \end{aligned}$$

Hence

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }\gtrsim & {} \Vert (uC_\varphi -K)f_{a,1}\Vert _{\mathcal {Z}} \ge \Vert uC_\varphi (f_{a,1}) \Vert _{\mathcal {Z}}-\Vert K(f_{a,1})\Vert _{\mathcal {Z}}, \end{aligned}$$

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }\gtrsim & {} \Vert (uC_\varphi -K)f_{a,2}\Vert _{\mathcal {Z}} \ge \Vert uC_\varphi (f_{a,2}) \Vert _{\mathcal {Z}}-\Vert K(f_{a,2})\Vert _{\mathcal {Z}} \end{aligned}$$

and

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }\gtrsim & {} \Vert (uC_\varphi -K)f_{a,3}\Vert _{\mathcal {Z}} \ge \Vert uC_\varphi (f_{a,3} )\Vert _{\mathcal {Z}}-\Vert K(f_{a,3})\Vert _{\mathcal {Z}}. \end{aligned}$$

Taking $\limsup _{|a|\rightarrow 1}$ to the last three inequalities on both sides, we obtain

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \gtrsim A, \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \gtrsim B , \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \gtrsim C . \end{aligned}$$

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

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }= \inf _{K} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \gtrsim \max \big \{ A, B , C\big \} . \end{aligned}$$

Next, we prove that

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \gtrsim \max \big \{ E, F \big \} . \end{aligned}$$

Let $\{z_j\}_{j\in {\mathbb {N}}}$ be a sequence in ${\mathbb {D}}$ such that $|\varphi (z_j)|\rightarrow 1$ as $j\rightarrow \infty$. Define

$$\begin{aligned} k_j(z)=\frac{1-|\varphi (z_j)|^2}{\big (1-\overline{\varphi (z_j)}z\big )^{1-\alpha }}-2\frac{\big (1-|\varphi (z_j)|^2\big )^2}{\big (1-\overline{\varphi (z_j)}z\big )^{2-\alpha }}+ \frac{\big (1-|\varphi (z_j)|^2\big )^3}{\big (1-\overline{\varphi (z_j)}z\big )^{3-\alpha }} \end{aligned}$$

and

$$\begin{aligned} m_j(z)=\frac{1-|\varphi (z_j)|^2}{\big (1-\overline{\varphi (z_j)}z\big )^{1-\alpha }}-\frac{5-2\alpha }{3-\alpha }\frac{\big (1-|\varphi (z_j)|^2\big )^2}{\big (1-\overline{\varphi (z_j)}z\big )^{2-\alpha }}+ \frac{2-\alpha }{3-\alpha }\frac{\big (1-|\varphi (z_j)|^2\big )^3}{\big (1-\overline{\varphi (z_j)}z\big )^{3-\alpha }}. \end{aligned}$$

Similarly to the above we see that both $k_j$ and $m_j$ belong to $\mathrm{Lip}_\alpha$ and converge to zero uniformly on compact subsets of ${\mathbb {D}}$ . Moreover,

$$\begin{aligned} k_j(\varphi (z_j))=0,~ k'_j(\varphi (z_j))=0,\quad \big |k''_j(\varphi (z_j))\big | =\frac{2|\varphi (z_j)|^2}{\left( 1-|\varphi (z_j)|^2\right) ^{2-\alpha }} \end{aligned}$$

and

$$\begin{aligned} m_j(\varphi (z_j))=0,~ m''_j(\varphi (z_j))=0,~|m'_j(\varphi (z_j))|=\frac{1}{3-\alpha } \frac{|\varphi (z_j)|}{\left( 1-|\varphi (z_j)|^2\right) ^{1-\alpha }}. \end{aligned}$$

Then for any compact operator $K: \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} ,$ we obtain

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }\gtrsim & {} \Vert (uC_\varphi -K)k_j\Vert _{\mathcal {Z}} \ge \Vert uC_\varphi ( k_j) \Vert _{\mathcal {Z}}-\Vert K (k_j)\Vert _{\mathcal {Z}} \end{aligned}$$

and

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }\gtrsim & {} \Vert (uC_\varphi -K)m_j\Vert _{\mathcal {Z}} \ge \Vert uC_\varphi (m_j) \Vert _{\mathcal {Z}}-\Vert K (m_j)\Vert _{\mathcal {Z}}. \end{aligned}$$

Taking the limit as $j \rightarrow \infty$ to the last two inequalities on both sides we get

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }\gtrsim & {} \limsup _{j \rightarrow \infty } \Vert uC_\varphi (k_j) \Vert _{\mathcal {Z}} \\\gtrsim & {} \limsup _{j \rightarrow \infty } \frac{\big (1-|z_j|^{2}\big ) |u(z_j)||\varphi '(z_j)|^2|\varphi (z_j)|^2}{\left( 1-|\varphi (z_j)|^{2}\right) ^{2-\alpha } } \\= & {} \limsup _{|\varphi (z)|\rightarrow 1}\frac{\big (1-|z|^{2}\big ) |u(z)||\varphi '(z)|^2}{\left( 1-|\varphi (z)|^{2}\right) ^{2-\alpha } }=E \end{aligned}$$

and

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }\gtrsim & {} \limsup _{j \rightarrow \infty } \Vert uC_\varphi (m_j) \Vert _{\mathcal {Z}} \\\gtrsim & {} \limsup _{j \rightarrow \infty } \frac{(1-|z_j|^{2}) |2u'(z_j)\varphi '(z_j)+u(z)\varphi ''(z_j)||\varphi (z_j)|}{\left( 1-|\varphi (z_j)|^{2}\right) ^{1-\alpha } } \\= & {} \limsup _{|\varphi (z)|\rightarrow 1}\frac{(1-|z|^{2}) |2u'(z)\varphi '(z)+u(z)\varphi ''(z)|}{\left( 1-|\varphi (z)|^{2}\right) ^{1-\alpha } }=F. \end{aligned}$$

By the definition of the essential norm, we obtain

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }= \inf _{K} \Vert uC_\varphi -K\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }\gtrsim \max \big \{ E, F \big \} . \end{aligned}$$

The upper estimate

For $r\in [0,1)$, set $K_r: H({\mathbb D})\rightarrow H({\mathbb D})$ by

$$\begin{aligned} (K_r f)(z)=f_r(z)=f(rz), ~~f\in H({\mathbb D}). \end{aligned}$$

It is obvious that $f_r \rightarrow f$ uniformly on compact subsets of ${\mathbb {D}}$ as $r \rightarrow 1$. Moreover, the operator $K_r$ is compact on $\mathrm{Lip}_\alpha$ and $\Vert K_r\Vert _{ \mathrm{Lip}_\alpha \rightarrow \mathrm{Lip}_\alpha }\le 1.$ Let $\{r_j\}\subset (0,1)$ be a sequence such that $r_j\rightarrow 1$ as $j\rightarrow \infty$. Then for all positive integer j, the operator $uC_\varphi K_{r_j}: \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$ is compact. By the definition of the essential norm we have

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \le \limsup _{j\rightarrow \infty }\Vert uC_\varphi - uC_\varphi K_{r_j}\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} }. \end{aligned}$$

(2.1)

Thus, we only need to show that

$$\begin{aligned}&\limsup _{j\rightarrow \infty }\Vert uC_\varphi -uC_\varphi K_{r_j}\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \lesssim \max \big \{ A, B, C \big \} \end{aligned}$$

and

$$\begin{aligned}&\limsup _{j\rightarrow \infty }\Vert uC_\varphi -uC_\varphi K_{r_j}\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \lesssim \max \big \{ E, F \big \}. \end{aligned}$$

For any $f\in \mathrm{Lip}_\alpha$ such that $\Vert f\Vert _{ \mathrm{Lip}_\alpha }\le 1$, we consider

$$\begin{aligned}&\Vert ( uC_\varphi - uC_\varphi K_{r_j})f\Vert _{\mathcal {Z}} \nonumber \\&=|u(0)f(\varphi (0))-u(0)f(r_j\varphi (0))|+\Vert u\cdot (f-f_{r_j})\circ \varphi \Vert _\beta \nonumber \\&\quad +|u'(0)(f-f_{r_j})(\varphi (0))+u(0)(f-f_{r_j})'(\varphi (0))\varphi '(0)|. \end{aligned}$$

(2.2)

It is clear that

$$\begin{aligned} \lim _{j\rightarrow \infty }|u(0)f(\varphi (0))-u(0)f(r_j\varphi (0))|=0 \end{aligned}$$

(2.3)

and

$$\begin{aligned} \lim _{j\rightarrow \infty }|u'(0)(f-f_{r_j})(\varphi (0))+u(0)(f-f_{r_j})'(\varphi (0))\varphi '(0)|=0. \end{aligned}$$

(2.4)

Now we consider

$$\begin{aligned}&\limsup _{j\rightarrow \infty }\Vert u\cdot (f-f_{r_j}) \circ \varphi \Vert _\beta \nonumber \\&\le \limsup _{j\rightarrow \infty }\sup _{z\in {\mathbb D}}(1-|z|^2) |(f-f_{r_j})(\varphi (z))||u''(z)|\nonumber \\&\quad +\limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2) |(f-f_{r_j})'(\varphi (z))||2u'(z)\varphi '(z)+u(z)\varphi ''(z)| \nonumber \\&\quad + \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2) |(f-f_{r_j})'(\varphi (z))||2u'(z)\varphi '(z)+u(z)\varphi ''(z)|\nonumber \\&\quad +\limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2) |(f-f_{r_j})''(\varphi (z))||\varphi '(z)|^2|u(z)| \nonumber \\&\quad + \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2) |(f-f_{r_j})''(\varphi (z))||\varphi '(z)|^2|u(z)|\nonumber \\&=Q_1+Q_2+Q_3+Q_4+Q_5, \end{aligned}$$

(2.5)

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

$$\begin{aligned} Q_1:= & {} \limsup _{j\rightarrow \infty }\sup _{z\in {\mathbb D}}(1-|z|^2) |(f-f_{r_j})(\varphi (z))||u''(z)| ,\\ Q_2:= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2) |(f-f_{r_j})'(\varphi (z))||2u'(z)\varphi '(z)+u(z)\varphi ''(z)|,\\ Q_3:= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2) |(f-f_{r_j})'(\varphi (z))||2u'(z)\varphi '(z)+u(z)\varphi ''(z)|,\\ Q_4:= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2) |(f-f_{r_j})''(\varphi (z))||\varphi '(z)|^2|u(z)| \end{aligned}$$

and

$$\begin{aligned} Q_5:= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}\big (1-|z|^2\big ) |\big (f-f_{r_j}\big )''(\varphi (z))||\varphi '(z)|^2|u(z)|. \end{aligned}$$

Since $uC_\varphi : \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$ is bounded, applying the operator $uC_{\varphi }$ to 1, z, and $z^2$, we easily get that $u\in {\mathcal {Z}} , u \varphi \in {\mathcal {Z}}$ and $u \varphi ^2 \in {\mathcal {Z}} ,$ i.e.,

$$\begin{aligned}&\sup _{z\in {\mathbb {D}}}|u''(z)|\big (1-|z|^2\big )<\infty ,\\&\sup _{z\in {\mathbb {D}}}|(u\varphi )''(z)|\big (1-|z|^2\big ) =\sup _{z\in {\mathbb {D}}}|2u'(z)\varphi '(z)+u(z)\varphi ''(z)\\&+\; u''(z)\varphi (z)|\big (1-|z|^2\big ) <\infty \end{aligned}$$

and

$$\begin{aligned}&\sup _{z\in {\mathbb {D}}}|\big (u\varphi ^2\big )''(z)|\big (1-|z|^2\big ) \\&=\sup _{z\in {\mathbb {D}}}\big |u''(z)\varphi ^2(z)+4u'(z)\varphi '(z)\varphi (z)+2u(z)\varphi ''(z)\varphi (z)\\&\quad +\, 2u(z)(\varphi '(z))^2\big |\big (1-|z|^2\big ) <\infty . \end{aligned}$$

Using the boundedness of $\varphi$, the last three inequalities and the triangle inequality, we get

$$\begin{aligned} \widetilde{K_1}:=\sup _{z\in {\mathbb D}}\big (1-|z|^2\big ) |2u'(z)\varphi '(z)+u(z)\varphi ''(z)|<\infty \end{aligned}$$

and

$$\begin{aligned} \widetilde{K_2}:=\sup _{z\in {\mathbb D}}\big (1-|z|^2\big ) |\varphi '(z)|^2|u(z)|<\infty . \end{aligned}$$

Since $r_jf'_{r_j}\rightarrow f',$ as well as $r^2_jf''_{r_j}\rightarrow f''$ uniformly on compact subsets of ${\mathbb D}$ as $j\rightarrow \infty$, we have

$$\begin{aligned} Q_2 \le \widetilde{ K_1} \limsup _{j\rightarrow \infty }\sup _{|w|\le r_N}|f'(w)- r_jf' (r_j w)| =0 \end{aligned}$$

(2.6)

and

$$\begin{aligned} Q_4 \le \widetilde{ K_2} \limsup _{j\rightarrow \infty }\sup _{|w|\le r_N}|f''(w)- r^2_jf'' (r_j w)| =0. \end{aligned}$$

(2.7)

By Lemma 2.2, from the fact that $u \in {\mathcal {Z}}$ and $f_{r_j}\rightarrow f$ uniformly on compact subsets of ${\mathbb D}$ as $j\rightarrow \infty$, we have

$$\begin{aligned} Q_1 \le \Vert u\Vert _{\mathcal {Z}}\limsup _{j\rightarrow \infty }\sup _{w\in {\mathbb D}}|f(w)- f(r_j w)| =0. \end{aligned}$$

(2.8)

For $Q_3$, we have $Q_3\le \limsup _{j\rightarrow \infty }(S_3+S_4),$ where

$$\begin{aligned} S_3:=\sup _{|\varphi (z)|>r_N}\big (1-|z|^2\big ) |f'(\varphi (z))||2u'(z)\varphi '(z)+u(z)\varphi ''(z)| \end{aligned}$$

and

$$\begin{aligned} S_4:=\sup _{|\varphi (z)|>r_N}\big (1-|z|^2\big ) r_j|f'(r_j\varphi (z))||2u'(z)\varphi '(z)+u(z)\varphi ''(z)|. \end{aligned}$$

Using the fact that $\Vert f\Vert _{ \mathrm{Lip}_\alpha }\le 1$, we have

$$\begin{aligned} S_3= & {} \sup _{|\varphi (z) |>r_N}\big (1-|z|^2\big ) |f'(\varphi (z))||2u'(z)\varphi '(z)+u(z)\varphi ''(z)| \\\lesssim & {} \frac{1}{r_N} \Vert f\Vert _{ \mathrm{Lip}_\alpha } \sup _{|\varphi (z) |>r_N} \big (1-|z|^2\big ) |2u'(z)\varphi '(z)+u(z)\varphi ''(z)|\frac{|\varphi (z)|}{\left( 1-|\varphi (z)|^2\right) ^{1-\alpha }} \\\lesssim & {} \sup _{|\varphi (z) |>r_N}\frac{1}{3-\alpha } \big (1-|z|^2\big ) |2u'(z)\varphi '(z)+u(z)\varphi ''(z)|\frac{|\varphi (z)|}{\left( 1-|\varphi (z)|^2\right) ^{1-\alpha }} \\\lesssim & {} \sup _{|a|>r_N} \Vert uC_\varphi \Big ( f_{a,1}- \frac{5-2\alpha }{3-\alpha } f_{a,2}+ \frac{2-\alpha }{3-\alpha } f_{a,3}\Big ) \Vert _{\beta } \\\lesssim & {} \sup _{|a|>r_N} \left\| uC_\varphi (f_{a,1})\right\| _{\mathcal {Z} }+ \sup _{|a|>r_N} \left\| uC_\varphi (f_{a,2})\right\| _{\mathcal {Z}}+\sup _{|a|>r_N} \left\| uC_\varphi (f_{a,3})\right\| _{\mathcal {Z}}. \end{aligned}$$

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

$$\begin{aligned}&\limsup _{j\rightarrow \infty }S_3 \nonumber \\\lesssim & {} \limsup _{|\varphi (z)|\rightarrow 1 } \frac{\big (1-|z|^2\big ) |2u'(z)\varphi '(z)+u(z)\varphi ''(z)|}{\left( 1-|\varphi (z)|^2\right) ^{1-\alpha }} =F\nonumber \\\lesssim & {} \limsup _{|a|\rightarrow 1 }\left\| uC_\varphi (f_{a,1})\right\| _{\mathcal {Z}}+ \limsup _{|a|\rightarrow 1 }\left\| uC_\varphi (f_{a,2})\right\| _{\mathcal {Z}}+ \limsup _{|a|\rightarrow 1 }\left\| uC_\varphi (f_{a,3})\right\| _{\mathcal {Z}} \nonumber \\= & {} A+B+C.\nonumber \end{aligned}$$

Similarly, we have $\limsup _{j\rightarrow \infty }S_4 \lesssim F \lesssim A+ B+C,$ i.e., we get

$$\begin{aligned} Q_3 \lesssim F \lesssim A + B + C . \end{aligned}$$

(2.9)

Next we consider $Q_5$. We have $Q_5\le \limsup _{j\rightarrow \infty }(S_5+S_6),$ where

$$\begin{aligned} S_5:=\sup _{|\varphi (z)|>r_N}\big (1-|z|^2\big ) |f''(\varphi (z))||\varphi '(z)|^2|u(z)| \end{aligned}$$

and

$$\begin{aligned} S_6:=\sup _{|\varphi (z)|>r_N}(1-|z|^2) r^2_j|f''(r_j\varphi (z))||\varphi '(z)|^2|u(z)|. \end{aligned}$$

Using the fact that $\Vert f\Vert _{ \mathrm{Lip}_\alpha }\le 1$ and after a calculation, we have

$$\begin{aligned} S_5= & {} \sup _{|\varphi (z) |>r_N}(1-|z|^2) |f''(\varphi (z))||\varphi '(z)|^2|u(z)| \\\lesssim & {} \frac{1}{r^2_N} \Vert f\Vert _{ \mathrm{Lip}_\alpha } \sup _{|\varphi (z) |>r_N} (1-|z|^2) |\varphi '(z)|^2|u(z)|\frac{|\varphi (z)|^2}{\left( 1-|\varphi (z)|^2\right) ^{2-\alpha }} \\\lesssim & {} \sup _{|\varphi (z) |>r_N} 2 (1-|z|^2) |\varphi '(z)|^2|u(z)|\frac{|\varphi (z)|^2}{\left( 1-|\varphi (z)|^2\right) ^{2-\alpha }} \\\lesssim & {} \sup _{|a|>r_N} \Vert uC_\varphi ( f_{a,1}- 2 f_{a,2}+ f_{a,3} ) \Vert _{\beta } \\\lesssim & {} \sup _{|a|>r_N} \left\| uC_\varphi (f_{a,1})\right\| _{\mathcal {Z}}+ \sup _{|a|>r_N} \left\| uC_\varphi (f_{a,2})\right\| _{\mathcal {Z}}+\sup _{|a|>r_N} \left\| uC_\varphi (f_{a,3})\right\| _{\mathcal {Z}}. \end{aligned}$$

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

$$\begin{aligned}&\limsup _{j\rightarrow \infty }S_5 \\\lesssim & {} \limsup _{|\varphi (z)|\rightarrow 1 } \frac{(1-|z|^2) |\varphi '(z)|^2|u(z)|}{\left( 1-|\varphi (z)|^2\right) ^{2-\alpha }} =E \\\lesssim & {} \limsup _{|a|\rightarrow 1 }\left\| uC_\varphi (f_{a,1})\right\| _{\mathcal {Z}}+ \limsup _{|a|\rightarrow 1 }\left\| uC_\varphi (f_{a,2})\right\| _{\mathcal {Z}}+ \limsup _{|a|\rightarrow 1 }\left\| uC_\varphi (f_{a,3}) \right\| _{\mathcal {Z}} \\= & {} A+B+C. \end{aligned}$$

Similarly, we have $\limsup _{j\rightarrow \infty }S_6 \lesssim E \lesssim A+ B+C,$ i.e., we get

$$\begin{aligned} Q_5 \lesssim E \lesssim A + B + C . \end{aligned}$$

(2.10)

Hence, by (2.2)–(2.10) we get

$$\begin{aligned}&\limsup _{j\rightarrow \infty }\Vert uC_\varphi - uC_\varphi K_{r_j}\Vert _{ \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \nonumber \\= & {} \limsup _{j\rightarrow \infty }\sup _{ \Vert f\Vert _{ \mathrm{Lip}_\alpha } \le 1}\Vert (uC_\varphi - uC_\varphi K_{r_j})f\Vert _{\mathcal {Z}} \nonumber \\= & {} \limsup _{j\rightarrow \infty }\sup _{ \Vert f\Vert _{ \mathrm{Lip}_\alpha } \le 1} \Vert u\cdot (f-f_{r_j}) \circ \varphi \Vert _{\beta }\nonumber \\\lesssim & {} E+F \lesssim A + B +C . \end{aligned}$$

(2.11)

Therefore, by (2.1) and (2.11), we obtain

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \lesssim E+F \lesssim \max \big \{ E, F \big \} \nonumber \end{aligned}$$

and

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \lesssim A+B+C \lesssim \max \big \{ A, B, C \big \} . \end{aligned}$$

This completes the proof of this theorem. $\square$

Theorem 2.2

Let $0<\alpha <1,$ $u \in H({\mathbb {D}})$ and $\varphi \in S({\mathbb D})$ with $\Vert \varphi \Vert _\infty =1$ such that $uC_\varphi : \mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$ is bounded. Then

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,\mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \approx \limsup _{k\rightarrow \infty }\Vert k^{-\alpha } u\varphi ^k \Vert _{\mathcal {Z}}. \end{aligned}$$

Proof

First, we prove that

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,\mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \ge \limsup _{k\rightarrow \infty }\Vert k^{-\alpha } u\varphi ^k \Vert _{\mathcal {Z}}. \end{aligned}$$

Let k be any positive integer. Let $f_{k}(z)=k^{-\alpha } z^k$. Then $\Vert f_{k}\Vert _{\mathrm{Lip}_\alpha }\approx 1$ and $f_k$ uniformly converges to zero on compact subsets of ${\mathbb D}$. By Lemma 2.1, for any compact operator $K:\mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$, we have

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert K (f_k)\Vert _{\mathcal {Z}} =0. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert uC_\varphi -K\Vert \ge \limsup _{k\rightarrow \infty }\Vert (uC_\varphi -K)f_k\Vert _ {\mathcal {Z}} \ge \limsup _{k\rightarrow \infty }\Vert uC_\varphi (f_k)\Vert _{\mathcal {Z}}. \end{aligned}$$

Therefore, by the definition of essential norm we get

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,\mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \ge \limsup _{k\rightarrow \infty }\Vert uC_\varphi (f_k)\Vert _{\mathcal {Z}} = \limsup _{k\rightarrow \infty }\Vert k^{-\alpha } u\varphi ^k \Vert _{\mathcal {Z}}. \end{aligned}$$

(2.12)

Next, we prove that

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,\mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \lesssim \limsup _{k\rightarrow \infty }\Vert k^{-\alpha } u\varphi ^k \Vert _{\mathcal {Z}}. \end{aligned}$$

Since $uC_\varphi :\mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}}$ is bounded, by Theorem 1 of [4] we see that

$$\begin{aligned} P:=\sup _{k \ge 0} \Vert k^{-\alpha } u\varphi ^{k} \Vert _ {\mathcal {Z}} <\infty . \end{aligned}$$

Consider the Maclaurin expansion of $f_{a, j}$,

$$\begin{aligned} f_{a,j}(z)= (1-|a|^2)^j\sum _{k=0}^{\infty } \frac{\Gamma (k+j-\alpha )}{\Gamma (j-\alpha )k!}\bar{a}^kz^{k}, ~~~ z\in {\mathbb D}. \end{aligned}$$

By Stirling’s formula, we get

$$\begin{aligned} \frac{\Gamma (k+j-\alpha )}{\Gamma (j-\alpha )k!} \approx k^{j-\alpha -1},\ \ ~ k\rightarrow \infty . \end{aligned}$$

Note that (see [33])

$$\begin{aligned} \frac{1}{(1-|a|)^\beta } = \sum _{k=0}^{\infty }\frac{\Gamma (k+\beta )}{k!\Gamma (\beta )}|a|^k~\text{ and }~ \frac{\Gamma (k+\beta )}{k!} \approx k^{\beta -1},\ \ ~ k\rightarrow \infty , \end{aligned}$$

we have

$$\begin{aligned} \sum _{k=0}^{\infty }\frac{\Gamma (k+j-\alpha )}{\Gamma (j-\alpha )k!} k^{\alpha }|a|^{k} \approx \sum _{k=0}^{\infty }k^{j-1}|a|^{k}\approx \frac{1}{(1-|a|)^j}. \end{aligned}$$

For any fix positive integer $n\ge 2$, it follows from the linearity of $uC_\varphi$ and the triangle inequality that

$$\begin{aligned} \Vert uC_{\varphi } f_{a,j}\Vert _ {\mathcal {Z}}\le & {} (1-|a|^2)^j \sum _{k=0}^{\infty }\frac{\Gamma (k+j-\alpha )}{\Gamma (j-\alpha )k!} k^\alpha |a|^{k}\Vert k^{-\alpha } u\varphi ^{k} \Vert _ {\mathcal {Z}} \\= & {} (1-|a|^2)^j\sum _{k=0}^{n-1}\frac{\Gamma (k+j-\alpha )}{\Gamma (j-\alpha )k!} k^\alpha |a|^{k}\Vert k^{-\alpha }u\varphi ^{k}\Vert _ {\mathcal {Z}} \\&+(1-|a|^2)^j\sum _{k=n}^{\infty }\frac{\Gamma (k+j-\alpha )}{\Gamma (j-\alpha )k!} k^\alpha |a|^{k}\Vert k^{-\alpha }u\varphi ^{k}\Vert _ {\mathcal {Z}} \\\lesssim & {} P(1-|a|^2)^j+(1-|a|^2)^j\sum _{k=n}^{\infty }\frac{\Gamma (k+j-\alpha )}{\Gamma (j-\alpha )k!} k^\alpha |a|^{k}\Vert k^{-\alpha }u\varphi ^{k}\Vert _ {\mathcal {Z}} \\\lesssim & {} P(1-|a|^2)^j+ \sup _{k\ge n}\Vert k^{-\alpha } u\varphi ^{k}\Vert _ {\mathcal {Z}}, \quad j=1,2,3. \end{aligned}$$

Letting $|a|\rightarrow 1$ in the above inequality. We get

$$\begin{aligned} \limsup _{|a|\rightarrow 1}\Vert uC_{\varphi } f_{a,j}\Vert _ {\mathcal {Z}} \lesssim \sup _{k\ge n}\Vert k^{-\alpha }u\varphi ^{k} \Vert _ {\mathcal {Z}}, \end{aligned}$$

for any positive integer $n\ge 2$. Thus,

$$\begin{aligned} \limsup _{|a|\rightarrow 1}\Vert uC_{\varphi } f_{a,j} \Vert _{\mathcal {Z}} \lesssim \limsup _{k\rightarrow \infty }\Vert k^{-\alpha }u\varphi ^{k} \Vert _ {\mathcal {Z}},\quad j=1,2,3, \end{aligned}$$

which implies that

$$\begin{aligned} \max \big \{ A, B, C \big \}\lesssim \limsup _{k\rightarrow \infty }\Vert k^{-\alpha }u\varphi ^{k} \Vert _ {\mathcal {Z}}. \end{aligned}$$

Therefore, by Theorem 2.1 we obtain

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,\mathrm{Lip}_\alpha \rightarrow {\mathcal {Z}} } \lesssim \max \big \{ A, B, C \big \} \lesssim \limsup _{k\rightarrow \infty }\Vert k^{-\alpha }u\varphi ^k \Vert _{\mathcal {Z}}. \end{aligned}$$

(2.13)

By (2.12) and (2.13), we get the desired result. The proof is completed. $\square$