Weighted Composition Operators from Analytic Besov Spaces into the Bloch Space

Abstract

In this paper, we give some new essential norm estimates of weighted composition operators $uC_{\varphi }$ from analytic Besov spaces into the Bloch space, where u is a function analytic on the unit disk $\mathbb {D}$ and $\varphi$ is an analytic self-map of $\mathbb {D}$. Moreover, new characterizations for the boundedness, compactness and essential norm of weighted composition operators $uC_{\varphi }$ are obtained by the nth power of the symbol $\varphi$ and the Volterra operators $I_u$ and $J_u$.

1 Introduction

Let $\mathbb {D}$ be the open unit disk in the complex plane ${\mathbb C}$ and $H(\mathbb {D})$ be the space of all analytic functions on ${\mathbb D}$. Let $S({\mathbb D})$ denote the set of all analytic self-maps of $\mathbb {D}$. The Bloch space, denoted by $\mathcal {B}=\mathcal {B}({\mathbb D})$, is the space of all $f \in H({\mathbb D})$ such that

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

Under the norm $\Vert f\Vert _{\mathcal {B}}=|f(0)|+ \Vert f\Vert _\beta$, the Bloch space is a Banach space.

For $p\in (1, \infty ),$ the analytic Besov space $B_p$ is the set of all $f\in H(\mathbb {D})$ for which

$$\begin{aligned} b_p(f)^p:=\int _{\mathbb {D}}\left| f'(z)\right| ^p \left( 1-|z|^2\right) ^{p-2}dA(z)<\infty , \end{aligned}$$

where dA is the normalized area measure on $\mathbb {D}.$ The quantity $b_p$ is a seminorm and the Besov norm is defined by $\Vert f\Vert _{B_p}=|f(0)|+b_p(f).$ In particular, $B_2$ is the classical Dirichlet space with an equivalent norm. See [18] for more results of the analytic Besov space.

Let $u \in H(\mathbb {D})$ and $\varphi \in S({\mathbb D})$. For $f \in H(\mathbb {D})$, the composition operator $C_\varphi$ and the multiplication operator $M_u$ are defined by

$$\begin{aligned} (C_\varphi f)(z) = f(\varphi (z)) ~~~\text{ and }~~~~~(M_u f)(z)=u(z)f(z), \end{aligned}$$

respectively. The weighted composition operator $uC_\varphi$ is defined by

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

It is clear that the weighted composition operator $uC_\varphi$ is the generalization of $C_\varphi$ and $M_u$. A main problem concerning concrete operators (such as composition operator, multiplication operator, weighted composition operator, Toeplitz operator and Hankel operator) is to relate operator theoretic properties to their function theoretic properties of their symbols.

It is well known that $C_\varphi$ is bounded on $\mathcal {B}$ by the Schwarz-Pick lemma for any $\varphi \in S({\mathbb D})$. The compactness of $C_\varphi$ on $\mathcal {B}$ was studied in [10, 14, 16]. Wulan et al. [16] proved 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 [17] showed that $\Vert C_\varphi \Vert _{e,\mathcal {B}\rightarrow \mathcal {B}} = \frac{e}{2 } \limsup _{n\rightarrow \infty } \Vert \varphi ^n \Vert _{\mathcal {B}}.$ Ohno and Zhao [13] studied the boundedness and compactness of the weighted composition operator $u C_\varphi :\mathcal {B} \rightarrow \mathcal {B}$. The essential norm of the operator $u C_\varphi :\mathcal {B} \rightarrow \mathcal {B}$ was studied in [5, 9, 11]. For more results on composition operator and weighted composition operators mapping into the Bloch space, see [1,2,3, 5,6,7,8,9,10,11, 13, 15,16,17] and the related references therein.

In [2], the authors characterized the boundedness and compactness of weighted composition operator $uC_\varphi :B_p \rightarrow \mathcal {B}$. Among others, they proved that, under the assumption that $uC_\varphi :B_p\rightarrow \mathcal {B}$ is bounded, $uC_\varphi :B_p \rightarrow \mathcal {B}$ is compact if and only if $\lim _{|\varphi (z)|\rightarrow 1} \Vert u C_{\varphi } f_{\varphi (z)} \Vert _{\mathcal {B}}=0 ~~~\text{ and }~~~\lim _{n\rightarrow \infty }\Vert u \varphi ^n\Vert _{\mathcal {B}}=0,$ as well as if and only if

$$\begin{aligned} \lim _{|\varphi (z)|\rightarrow 1} \left( 1-|z|^{2}\right) \left| u'(z)\right| \left( \log \frac{2}{1-|\varphi (z)|^2}\right) ^{1-\frac{1}{p}} =0 \end{aligned}$$

and

$$\begin{aligned} \lim _{|\varphi (z)|\rightarrow 1}\frac{\left( 1-|z|^{2}\right) \left| u(z)\varphi '(z)\right| }{ 1-|\varphi (z)|^{2} }=0. \end{aligned}$$

Here

$$\begin{aligned} f_a(z)=\left( \log \frac{e}{1-|a|^2}\right) ^{-\frac{1}{p}} \log \frac{e}{1-\bar{a}z} . \end{aligned}$$

Motivated by the above result, in this paper, we give the corresponding estimates for the essential norm of the operator $uC_{\varphi }:B_p \rightarrow \mathcal {B}$. Moreover, we give a new characterization for the boundedness, compactness and essential norm for the operator $uC_{\varphi }:B_p \rightarrow \mathcal {B}$.

Recall that the essential norm $\Vert T\Vert _{e,X\rightarrow Y}$ of a bounded linear operator $T:X\rightarrow Y$ is defined as the distance from T to the set of compact operators K mapping X into Y, that is, $\Vert T\Vert _{e, X\rightarrow Y}=\inf \{\Vert T-K\Vert _{X\rightarrow Y}: K~ \text{ is } \text{ compact }~~\},$ where $\Vert \cdot \Vert _{X\rightarrow Y}$ is the operator norm.

Throughout this paper, 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 Essential Norm of $uC_\varphi :B_p \rightarrow \mathcal {B}$

In this section, we give some estimates for the essential norm of the operator $uC_\varphi :B_p \rightarrow \mathcal {B}$. For this purpose, we need some lemmas which will be used in the proofs of the main results in this paper.

Lemma 2.1

[14] 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

[2] Let $1<p<\infty$. If $f\in B_p,$ then

• (i) $|f(z)|\lesssim \Vert f\Vert _{B_p}\left( \log \frac{2}{1-|z|^2}\right) ^{1-\frac{1}{p}}$, for every $z\in \mathbb {D};$

• (ii) $|f'(z)|\lesssim \frac{1}{1-|z|^2}\Vert f\Vert _{B_p}$, for every $z\in \mathbb {D}.$

Let $a\in {\mathbb D}$. We define

$$\begin{aligned} g_a(z)=\frac{\left( \log \frac{e}{1-\bar{a}z}\right) ^2}{\left( \log \frac{e}{1-|a|^2}\right) ^{1+\frac{1}{p}}}, ~~~~~~~ h_a(z)=\frac{(a-z)\left( 1-|a|^2\right) }{(1-\bar{a}z)^2},~~z\in {\mathbb D}. \end{aligned}$$

We state and prove the first result in this section.

Theorem 2.1

Let $1< p< \infty , u \in H(\mathbb {D})$ and $\varphi \in S({\mathbb D})$ such that $uC_\varphi :B_p \rightarrow \mathcal {B}$ is bounded. Then

$$\begin{aligned} \Vert uC_{\varphi }\Vert _{e,B_p \rightarrow \mathcal {B} } \approx \max \Big \{ A, B, C \Big \} \approx \max \Big \{E, F \Big \}, \nonumber \end{aligned}$$

where

$$\begin{aligned}&A:=\limsup _{|a|\rightarrow 1} \Vert uC_{\varphi } f_a \Vert _{ \mathcal {B} },~~~~B:= \limsup _{|a|\rightarrow 1} \Vert uC_{\varphi }g_a \Vert _{ \mathcal {B} }, C:= \limsup _{|a|\rightarrow 1} \Vert uC_{\varphi }h_a \Vert _{ \mathcal {B} },\\&E:=\limsup _{ |\varphi (z)|\rightarrow 1}\left( 1-|z|^2\right) \left| u'(z)\right| \left( \log \frac{e}{1-|\varphi (z)|^2}\right) ^{1-\frac{1}{p}} \end{aligned}$$

and

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

Proof

Without loss of generality, we assume that $\Vert \varphi \Vert _\infty =1$. We first prove that

$$\begin{aligned} \max \Big \{ A, B, C \Big \} \lesssim \Vert uC_\varphi \Vert _{e,B_p \rightarrow \mathcal {B} } . \end{aligned}$$

(2.1)

As shown in [2], $f_a, g_a, h_a\in B_p, \Vert f_a\Vert _{B_p}, \Vert g_a\Vert _{B_p}, \Vert h_a\Vert _{B_p}$ are bounded by a constant independent of a, and the all $f_a, g_a$ and $h_a$ converge to zero uniformly on compact subsets of ${\mathbb D}$ as $|a|\rightarrow 1$. Thus, for any compact operator $K:B_p \rightarrow \mathcal {B}$, by Lemma 2.1 we have

$$\begin{aligned} \lim _{|a|\rightarrow 1}\Vert Kf_a\Vert _{\mathcal {B} }=0, \lim _{|a|\rightarrow 1}\Vert Kg_a\Vert _{\mathcal {B} }=0, \lim _{|a|\rightarrow 1}\Vert Kh_a\Vert _{\mathcal {B} }=0. \end{aligned}$$

Since $uC_\varphi :B_p \rightarrow \mathcal {B}$ is bounded, we have

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B}}\gtrsim & {} \Vert (uC_\varphi -K)f_a\Vert _{\mathcal {B} } \ge \Vert uC_\varphi (f_a) \Vert _{\mathcal {B} }-\Vert Kf_a\Vert _{\mathcal {B} } ,\nonumber \\ \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B}}\gtrsim & {} \Vert (uC_\varphi -K)g_a\Vert _{\mathcal {B} } \ge \Vert uC_\varphi (g_a) \Vert _{\mathcal {B} }-\Vert Kg_a\Vert _{\mathcal {B} } \nonumber \end{aligned}$$

and

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B}}\gtrsim & {} \Vert (uC_\varphi -K)h_a\Vert _{\mathcal {B} } \ge \Vert uC_\varphi (h_a) \Vert _{\mathcal {B} }-\Vert Kh_a\Vert _{\mathcal {B} } .\nonumber \end{aligned}$$

Thus,

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B}} \gtrsim A,~~~~~~ \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B}} \gtrsim B,~~~~~\Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B}} \gtrsim C .\nonumber \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,B_p \rightarrow \mathcal {B} }= \inf _{K} \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B} } \gtrsim \max \Big \{ A, B, C \Big \} .\nonumber \end{aligned}$$

Next, we prove that

$$\begin{aligned} \max \Big \{ E, F \Big \} \lesssim \Vert uC_\varphi \Vert _{e,B_p \rightarrow \mathcal {B} }. \end{aligned}$$

(2.2)

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

$$\begin{aligned} k_j(z)=\frac{\log \frac{e}{1-\overline{\varphi (z_j)}z}}{\left( \log \frac{e}{1-|\varphi (z_j)|^2}\right) ^{\frac{1}{p}}}-\frac{1}{2} \frac{\left( \log \frac{e}{1-\overline{\varphi (z_j)}z}\right) ^2}{\left( \log \frac{e}{1-|\varphi (z_j)|^2}\right) ^{1+\frac{1}{p}}} \end{aligned}$$

and

$$\begin{aligned} l_j(z)=\frac{\left( \varphi (z_j)-z\right) \left( 1-|\varphi (z_j)|^2\right) }{\left( 1-\overline{\varphi (z_j)}z\right) ^2}. \end{aligned}$$

We know that the both $k_j$ and $l_j$ belong to $B_p$ and converge to zero uniformly on compact subsets of $\mathbb {D}$. Moreover,

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

and

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

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

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B}} \gtrsim \Vert (uC_\varphi -K) k_j \Vert _{\mathcal {B} }\gtrsim \Vert uC_\varphi (k_j) \Vert _{\mathcal {B} }- \Vert Kk_j\Vert _{\mathcal {B} } \end{aligned}$$

and

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B}} \gtrsim \Vert (uC_\varphi -K) l_j \Vert _{\mathcal {B} }\gtrsim \Vert uC_\varphi (l_j) \Vert _{\mathcal {B} }- \Vert Kl_j\Vert _{\mathcal {B} }. \end{aligned}$$

Taking $j\rightarrow \infty$ , we get

$$\begin{aligned} \Vert uC_\varphi -K\Vert _{B_p \rightarrow \mathcal {B} }\gtrsim & {} \limsup _{j\rightarrow \infty } \Vert uC_\varphi (k_j) \Vert _{ \mathcal {B} } \\\gtrsim & {} \limsup _{j\rightarrow \infty } \left( 1-|z_j|^{2}\right) \left| u'(z_j)\right| \left( \log \frac{e}{1-|\varphi (z_j)|^2}\right) ^{1-\frac{1}{p}}\\= & {} \limsup _{|\varphi (z)|\rightarrow 1}\left( 1-|z|^{2}\right) \left| u'(z)\right| \left( \log \frac{e}{1-|\varphi (z)|^2}\right) ^{1-\frac{1}{p}}=E \end{aligned}$$

and

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

Hence, we obtain (2.2).

Now, we show that

$$\begin{aligned} ~~~~\Vert uC_\varphi \Vert _{e,B_p \rightarrow \mathcal {B} } \lesssim \max \Big \{ A, B , C\Big \}~~~~ \text{ and }~~~~ \Vert uC_\varphi \Vert _{e,B_p \rightarrow \mathcal {B} } \lesssim \max \Big \{ E, F \Big \}. \qquad \end{aligned}$$

(2.3)

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

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,B_p \rightarrow \mathcal {B} } \le \limsup _{j\rightarrow \infty }\Vert uC_\varphi - uC_\varphi K_{r_j}\Vert _{B_p \rightarrow \mathcal {B} }.\nonumber \end{aligned}$$

To give (2.3), we only need to show that

$$\begin{aligned}&\limsup _{j\rightarrow \infty }\Vert uC_\varphi -uC_\varphi K_{r_j}\Vert _{ B_p \rightarrow \mathcal {B} } \lesssim \max \Big \{ A, B, C \Big \} \end{aligned}$$

(2.4)

and

$$\begin{aligned}&\limsup _{j\rightarrow \infty }\Vert uC_\varphi -uC_\varphi K_{r_j}\Vert _{ B_p \rightarrow \mathcal {B} } \lesssim \max \Big \{ E, F \Big \}. \end{aligned}$$

(2.5)

For $f\in B_p$ with $\Vert f\Vert _{B_p}\le 1$, we consider

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

It is obvious that $\lim _{j\rightarrow \infty }|u(0)f(\varphi (0))-u(0)f_{r_j}(\varphi (0))|=0.$ Let $N\in \mathbb {N }$ be large enough such that $r_j\ge \frac{1}{2}$ for all $j\ge N$ and we have

$$\begin{aligned} \limsup _{j\rightarrow \infty }\Vert u\cdot \left( f-f_{r_j}\right) \circ \varphi \Vert _\beta \lesssim Q_1+Q_2+Q_3+Q_4, \end{aligned}$$

(2.6)

where

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

and

$$\begin{aligned} Q_4:=\limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}\left( 1-|z|^2\right) \left| \left( f-f_{r_j}\right) (\varphi (z))||u'(z)\right| . \end{aligned}$$

Since $uC_\varphi :B_p \rightarrow \mathcal {B}$ is bounded, from [2] we see that $u\in \mathcal {B}$ and

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

Since $f'_{r_j}\rightarrow f'$ uniformly on compact subsets of ${\mathbb D}$ as $j\rightarrow \infty$, we have

$$\begin{aligned} Q_1 \le \widetilde{Q} \limsup _{j\rightarrow \infty }\sup _{|w|\le r_N}\left| f'(w)- f_{r_j}' ( w)\right| =0. \end{aligned}$$

(2.7)

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

$$\begin{aligned} Q_3 \le \Vert u\Vert _{ \mathcal {B} }\limsup _{j\rightarrow \infty }\sup _{|w|\le r_N}\left| f(w)- f(r_j w)\right| =0. \end{aligned}$$

(2.8)

It is clear that $Q_2=\limsup _{j\rightarrow \infty }Q_{21},$ where

$$\begin{aligned} Q_{21}:=\sup _{|\varphi (z)|> r_N}\left( 1-|z|^2\right) \left| \left( f-f_{r_j}\right) '(\varphi (z))||\varphi '(z)||u(z)\right| .\nonumber \end{aligned}$$

Using Lemma 2.2 and the fact that $\Vert f\Vert _{B_p}\le 1$, we have

$$\begin{aligned} Q_{21}= & {} \sup _{|\varphi (z) |>r_N}\left( 1-|z|^2\right) \left| \left( f-f_{r_j}\right) '(\varphi (z))| |\varphi '(z)||u(z)\right| \nonumber \\\lesssim & {} \Vert f-f_{r_j}\Vert _{B_p}\sup _{|\varphi (z) |>r_N} \frac{ \left( 1-|z|^2\right) |\varphi '(z)||u(z)| }{1-|\varphi (z)|^2 } \nonumber \\\lesssim & {} \sup _{|\varphi (z)|>r_N}\left( 1-|z|^2\right) \big |u'(z)\frac{(\varphi (z)-\varphi (z))(1-|\varphi (z)|^2)}{(1-|\varphi (z)|^2)^2}\nonumber \\&+\,u(z)\varphi '(z)\frac{(1-|\varphi (z)|^2)(2|\varphi (z)|^2-1-|\varphi (z)|^2)}{(1-|\varphi (z)|^2)^3}\big |\nonumber \\\lesssim & {} \sup _{|\varphi (z)|>r_N}\sup _{|a|>r_N}\left( 1-|z|^2\right) \big |u'(z)\frac{(a-\varphi (z))\left( 1-|a|^2\right) }{(1-\bar{a}\varphi (z))^2}\nonumber \\&+\,u(z)\varphi '(z)\frac{\left( 1-|a|^2\right) (2|a|^2-1-\bar{a}\varphi (z))}{(1-\bar{a}\varphi (z))^3}\big |\nonumber \\= & {} \sup _{|\varphi (z)|>r_N}\sup _{|a|>r_N}\left( 1-|z|^2\right) |(uC_\varphi h_a)'(z)|\nonumber \\\lesssim & {} \sup _{|a|>r_N} \left\| uC_\varphi (h_a)\right\| _{ \mathcal {B} } .\nonumber \end{aligned}$$

Letting $N\rightarrow \infty$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }Q_{21}\lesssim & {} \limsup _{|\varphi (z)|\rightarrow 1} \frac{ \left( 1-|z|^2\right) |\varphi '(z)||u(z)| }{1-|\varphi (z)|^2 }(=F) \lesssim \limsup _{|a|\rightarrow 1}\left\| uC_\varphi (h_a)\right\| _{ \mathcal {B} }.\nonumber \end{aligned}$$

Hence,

$$\begin{aligned} Q_2\lesssim F\lesssim C . \end{aligned}$$

(2.9)

We know that $Q_4=\limsup _{j\rightarrow \infty }Q_{41},$ where

$$\begin{aligned} Q_{41}:=\sup _{|\varphi (z)|> r_N}\left( 1-|z|^2\right) |\left( f-f_{r_j}\right) (\varphi (z))||u'(z)|.\nonumber \end{aligned}$$

Using a similar estimates to $Q_{21}$, Lemma 2.2 and the fact that $\Vert f\Vert _{B_p}\le 1$, we get

$$\begin{aligned} Q_{41}= & {} \sup _{|\varphi (z)|>r_N}\left( 1-|z|^2\right) |\left( f-f_{r_j}\right) (\varphi (z))||u'(z)|\nonumber \\\lesssim & {} \frac{1}{2} \sup _{|\varphi (z) |>r_N} \left( 1-|z|^2\right) |u'(z)|\left( \log \frac{e}{1-|\varphi (z)|^2}\right) ^{1-\frac{1}{p}} \nonumber \\\lesssim & {} \sup _{|a|>r_N} \left\| uC_\varphi \left( f_a- \frac{1}{2} g_a \right) \right\| _{ \mathcal {B} } \lesssim \sup _{|a|>r_N} \left\| uC_\varphi (f_a)\right\| _{ \mathcal {B} }+ \sup _{|a|>r_N} \left\| uC_\varphi (g_a)\right\| _{ \mathcal {B} } .\nonumber \end{aligned}$$

Taking $N\rightarrow \infty$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }Q_{41}\lesssim & {} \limsup _{|\varphi (z)|\rightarrow 1 } \left( 1-|z|^2\right) |u'(z)|\left( \log \frac{e}{1-|\varphi (z)|^2}\right) ^{1-\frac{1}{p}} ( =E)\nonumber \\\lesssim & {} \limsup _{|a|\rightarrow 1 }\left\| uC_\varphi (f_a)\right\| _{ \mathcal {B} }+ \limsup _{|a|\rightarrow 1 }\left\| uC_\varphi (g_a)\right\| _{ \mathcal {B} }= A+B, \nonumber \end{aligned}$$

and then

$$\begin{aligned} Q_4 \lesssim E \lesssim A + B. \end{aligned}$$

(2.10)

Thus, by the above estimates (2.6)–(2.10) we get (2.4) and (2.5). Therefore,

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,B_p \rightarrow \mathcal {B} } \lesssim E+F \lesssim \max \Big \{ E, F \Big \} \nonumber \end{aligned}$$

and

$$\begin{aligned} \Vert uC_\varphi \Vert _{e,B_p \rightarrow \mathcal {B} } \lesssim A+B+C \lesssim \max \Big \{ A, B, C \Big \} . \nonumber \end{aligned}$$

Hence, by (2.1)–(2.3) we get the desired result. This completes the proof of this theorem.$\square$

Theorem 2.2

Let $1< p< \infty , u \in H(\mathbb {D})$ and $\varphi \in S({\mathbb D})$ such that $uC_\varphi :B_p \rightarrow \mathcal {B}$ is bounded. Then

$$\begin{aligned} \Vert uC_{\varphi }\Vert _{e,B_p \rightarrow \mathcal {B} } \approx \max \Big \{ \limsup _{|a|\rightarrow 1}\big \Vert uC_{\varphi }f_a \big \Vert _{\mathcal {B}}, \limsup _{n\rightarrow \infty } \Vert u\varphi ^n\Vert _{\mathcal {B}} \Big \} . \nonumber \end{aligned}$$

Proof

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

$$\begin{aligned} q_{a}(z)=\frac{1-|a|^2}{1-\overline{a}z}. \end{aligned}$$

It is easy to check that the both $f_{\varphi (z_j)}$ and $q_{\varphi (z_j)}$ belong to $B_p$ and converge to zero uniformly on compact subsets of $\mathbb {D}$. Here, $f_a$ is defined in the proof of Theorem 2.1. Since

$$\begin{aligned} \Vert uC_{\varphi }f_{\varphi (z_j)}\Vert _{\mathcal {B}}\ge & {} \sup _{z\in \mathbb {D}}\left( 1-|z|^2\right) |u(z)f'_{\varphi (z_j)}\left( \varphi (z)\right) \varphi '(z)+u'(z)f_{\varphi (z_j)}(\varphi (z))|\\\ge & {} \left( 1-|z_j|^2\right) |u'(z_j)|\left( \log \frac{e}{1-|\varphi (z_j)|^2}\right) ^{1-\frac{1}{p}}\\&-\frac{\left( 1-|z_j|^2\right) |u(z_j)\overline{\varphi (z_j)}\varphi '(z_j)|}{1-|\varphi (z_j)|^2}\left( \log \frac{e}{1-|\varphi (z_j)|^2}\right) ^{-\frac{1}{p}} \end{aligned}$$

and

$$\begin{aligned} \Vert uC_{\varphi }q_{\varphi (z_j)}\Vert _{\mathcal {B}}\ge & {} \sup _{z\in \mathbb {D}}\left( 1-|z|^2\right) |u(z)q'_{\varphi (z_j)}(\varphi (z))\varphi '(z)+u'(z)q_{\varphi (z_j)}(\varphi (z))|\\\ge & {} \frac{\left( 1-|z_j|^2)|u(z_j\right) \overline{\varphi (z_j)}\varphi '(z_j)|}{1-|\varphi (z_j)|^2}-\left( 1-|z_j|^2\right) |u'(z_j)|. \end{aligned}$$

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

$$\begin{aligned}&\limsup _{|\varphi (z)|\rightarrow 1}\Vert uC_{\varphi }f_{\varphi (z)}\Vert _{\mathcal {B}}+\limsup _{|\varphi (z)|\rightarrow 1}\frac{\left( 1-|z|^2\right) |u(z)||\varphi '(z)|}{1-|\varphi (z)|^2}\left( \log \frac{e}{1-|\varphi (z)|^2}\right) ^{-\frac{1}{p}}\\&\quad =\limsup _{j\rightarrow \infty }\Vert uC_{\varphi }f_{\varphi (z_j)}\Vert _{\mathcal {B}}\\&\qquad +\limsup _{j\rightarrow \infty }\frac{\left( 1-|z_j|^2\right) |u(z_j)\overline{\varphi (z_j)}\varphi '(z_j)|}{1-|\varphi (z_j)|^2}\left( \log \frac{e}{1-|\varphi (z_j)|^2}\right) ^{-\frac{1}{p}}\\&\quad \ge \limsup _{j\rightarrow \infty }\left( 1-|z_j|^2\right) |u'(z_j)|\left( \log \frac{e}{1-|\varphi (z_j)|^2}\right) ^{1-\frac{1}{p}}\\&\quad \ge \limsup _{j\rightarrow \infty }\left( 1-|z_j|^2\right) |u'(z_j)|=\limsup _{|\varphi (z)|\rightarrow 1}\left( 1-|z|^2\right) |u'(z)| \end{aligned}$$

and

$$\begin{aligned}&\limsup _{|\varphi (z)|\rightarrow 1}\Vert uC_{\varphi }q_{\varphi (z)}\Vert _{\mathcal {B}}+\limsup _{|\varphi (z)|\rightarrow 1}\left( 1-|z|^2\right) |u'(z)|\\&\quad =\limsup _{j\rightarrow \infty }\Vert uC_{\varphi }q_{\varphi (z_j)}\Vert _{\mathcal {B}}+\limsup _{j\rightarrow \infty }\left( 1-|z_j|^2\right) |u'(z_j)|\\&\quad \ge \limsup _{j\rightarrow \infty }\frac{\left( 1-|z_j|^2\right) |u(z_j)\overline{\varphi (z_j)}\varphi '(z_j)|}{1-|\varphi (z_j)|^2}\\&\quad =\limsup _{|\varphi (z)|\rightarrow 1}\frac{\left( 1-|z|^2\right) |u(z)||\varphi '(z)|}{1-|\varphi (z)|^2}. \end{aligned}$$

Since $uC_\varphi :B_p \rightarrow \mathcal {B}$ is bounded, we have (see [2])

$$\begin{aligned} \sup _{z\in {\mathbb D}}\frac{\left( 1-|z|^2\right) |u(z)||\varphi '(z)|}{1-|\varphi (z)|^2}<\infty , \end{aligned}$$

which implies that

$$\begin{aligned} \limsup _{|\varphi (z)|\rightarrow 1}\frac{\left( 1-|z|^2\right) |u(z)||\varphi '(z)|}{1-|\varphi (z)|^2}\left( \log \frac{e}{1-|\varphi (z)|^2}\right) ^{-\frac{1}{p}}=0. \end{aligned}$$

Thus, we have

$$\begin{aligned} \limsup _{|\varphi (z)|\rightarrow 1}\left( 1-|z|^2\right) |u'(z)|\left( \log \frac{e}{1-|\varphi (z)|^2}\right) ^{1-\frac{1}{p}}\le & {} \limsup _{|\varphi (z)|\rightarrow 1}\Vert uC_{\varphi }f_{\varphi (z)}\Vert _{\mathcal {B}}\nonumber \\\le & {} \limsup _{|a|\rightarrow 1}\Vert uC_{\varphi }f_{a}\Vert _{\mathcal {B}}, \qquad \end{aligned}$$

(2.11)

$\limsup _{|\varphi (z)|\rightarrow 1}\left( 1-|z|^2\right) |u'(z)| \le \limsup _{|a|\rightarrow 1}\Vert uC_{\varphi }f_{a}\Vert _{\mathcal {B}}$ and then

$$\begin{aligned} \limsup _{|\varphi (z)|\rightarrow 1}\frac{\left( 1-|z|^2\right) |u(z)||\varphi '(z)|}{1-|\varphi (z)|^2}\le & {} \limsup _{|\varphi (z)|\rightarrow 1}\Vert uC_{\varphi }q_{\varphi (z)}\Vert _{\mathcal {B}}+\limsup _{|\varphi (z)|\rightarrow 1}\Vert uC_{\varphi }f_{\varphi (z)}\Vert _{\mathcal {B}}\nonumber \\\le & {} \limsup _{|a|\rightarrow 1}\Vert uC_{\varphi }q_{a}\Vert _{\mathcal {B}}+\limsup _{|a|\rightarrow 1}\Vert uC_{\varphi }f_{a}\Vert _{\mathcal {B}}. \nonumber \\ \end{aligned}$$

(2.12)

By (2.11), (2.12) and Theorem 2.1, we obtain

$$\begin{aligned} \limsup _{|a|\rightarrow 1}\Vert uC_{\varphi }f_{a}\Vert _{\mathcal {B}}\le & {} \Vert uC_{\varphi }\Vert _{e,B_p \rightarrow \mathcal {B} } \approx \max \Big \{E, F \Big \} \nonumber \\\le & {} \max \Big \{\limsup _{|a|\rightarrow 1}\Vert uC_{\varphi }f_{a}\Vert _{\mathcal {B}}, \limsup _{|a|\rightarrow 1}\Vert uC_{\varphi }q_{a}\Vert _{\mathcal {B}} \Big \} .\qquad \end{aligned}$$

(2.13)

To finish the proof, we only need to prove that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert u\varphi ^n\Vert _{\mathcal {B}}\le \Vert uC_{\varphi }\Vert _{e,B_p \rightarrow \mathcal {B} } \end{aligned}$$

and

$$\begin{aligned} \limsup _{|a|\rightarrow 1}\Vert uC_{\varphi }q_{a}\Vert _{\mathcal {B}} \lesssim \limsup _{n\rightarrow \infty }\Vert u\varphi ^n\Vert _{\mathcal {B}}. \end{aligned}$$

For each nonnegative integer n, let $p_n(z)=z^n.$ Then, $p_n\in B_p$ and the sequence $\{p_n\}$ converges to zero uniformly on compact subsets of $\mathbb {D}$. Thus, for any compact operator $K: B_p \rightarrow \mathcal {B},$ by Lemma 2.1 we have $\lim _{n \rightarrow \infty } \Vert Kp_n\Vert _{\mathcal {B}}=0.$ Hence,

$$\begin{aligned} \Vert uC_{\varphi }-K\Vert _{B_p \rightarrow \mathcal {B}}\gtrsim & {} \limsup _{n \rightarrow \infty }\Vert (uC_{\varphi }-K)p_n\Vert _{\mathcal {B}} \ge \limsup _{n \rightarrow \infty }\Vert uC_{\varphi }p_n\Vert _{\mathcal {B}}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert uC_{\varphi }\Vert _{e, B_p \rightarrow \mathcal {B}}= & {} \inf _{K}\Vert uC_{\varphi }-K\Vert _{B_p \rightarrow \mathcal {B}} \ge \limsup _{n \rightarrow \infty } \Vert uC_{\varphi }p_n\Vert _{\mathcal {B}}\nonumber \\= & {} \limsup _{n \rightarrow \infty } \Vert u\varphi ^n\Vert _{\mathcal {B}}. \end{aligned}$$

(2.14)

On the other hand, let $a\in \mathbb {D}$, then

$$\begin{aligned} q_a(z)=\frac{1-|a|^2 }{1-\overline{a}z }=\left( 1-|a|^2\right) \sum _{k=0}^{\infty }\bar{a}^kz^k. \end{aligned}$$

For fixed positive integer $n\ge 1,$ it follows from the triangle inequality and the fact $\sup _{0\le k<\infty }\Vert u\varphi ^k\Vert _{\mathcal {B}}<\infty$ that

$$\begin{aligned} \Vert uC_{\varphi }q_a\Vert _{\mathcal {B}}\le & {} \left( 1-|a|^2\right) \sum _{k=0}^{\infty }|a|^k\Vert u\varphi ^k\Vert _{\mathcal {B}}\\= & {} \left( 1-|a|^2\right) \sum _{k=0}^{n-1}|a|^k\Vert u\varphi ^k\Vert _{\mathcal {B}}+\left( 1-|a|^2\right) \sum _{k=n}^{\infty }|a|^k\Vert u\varphi ^k\Vert _{\mathcal {B}}\\\le & {} n\left( 1-|a|^2\right) \sup _{0\le k\le n-1}\Vert u\varphi ^k\Vert _{\mathcal {B}}+\left( 1-|a|^2\right) \sum _{k=n}^{\infty }|a|^k\sup _{j\ge n}\Vert u\varphi ^j\Vert _{\mathcal {B}}\\\lesssim & {} n\left( 1-|a|^2\right) +2\sup _{k\ge n}\Vert u\varphi ^k\Vert _{\mathcal {B}}. \end{aligned}$$

Letting $|a| \rightarrow 1$ in the above inequality leads to

$$\begin{aligned} \limsup _{|a| \rightarrow 1}\Vert uC_{\varphi }q_a\Vert _{\mathcal {B}}\lesssim \sup _{k\ge n}\Vert u\varphi ^k\Vert _{\mathcal {B}} \end{aligned}$$

for any positive integer $n\ge 1.$ Thus

$$\begin{aligned} \limsup _{|a| \rightarrow 1}\Vert uC_{\varphi }q_a\Vert _{\mathcal {B}}\lesssim \limsup _{n \rightarrow \infty }\Vert u\varphi ^n\Vert _{\mathcal {B}}. \end{aligned}$$

(2.15)

Therefore, by (2.13) , (2.14) and (2.15) we get the desired result. This completes the proof of this theorem.$\square$

3 A New Characterization of $uC_\varphi : B_p \rightarrow \mathcal {B}$

In this section, we give a new characterization for the boundedness, compactness and essential norm of the operator $uC_\varphi : B_p \rightarrow \mathcal {B}$. For this purpose, we state some definitions and some lemmas which will be used.

Let $v:{\mathbb D}\rightarrow R_+$ be a continuous, strictly positive and bounded function. The weighted space $H^\infty _v$ is the space which consisting of all $f\in H({\mathbb D})$ such that $\Vert f\Vert _v=\sup _{z \in \mathbb {D}}v(z)|f(z)|<\infty .$$H^\infty _v$ is a Banach space under the norm $\Vert \cdot \Vert _v$. The weight v is called radial if $v(z)=v(|z|)$ for all $z\in {\mathbb D}$. The associated weight $\widetilde{v}$ of v is defined by (see [4])

$$\begin{aligned} \widetilde{v}=(\sup \{|f(z)|: f\in H^\infty _v, \Vert f\Vert _v \le 1 \})^{-1}, z\in {\mathbb D}. \end{aligned}$$

Let $0<\alpha <\infty$. When $v=v_\alpha (z)=\left( 1-|z|^2\right) ^\alpha$, it is easy to check that $\widetilde{v}_\alpha (z)=v_\alpha (z)$. In this case, we denote $H^\infty _v$ by $H^\infty _{v_\alpha }$. Here

$$\begin{aligned} H^\infty _{v_\alpha }=\left\{ f\in H({\mathbb D}) : \Vert f\Vert _{v_\alpha }=\sup _{z \in \mathbb {D}} |f(z)|\left( 1-|z|^2\right) ^\alpha <\infty \right\} . \end{aligned}$$

When

$$\begin{aligned} v=v_{\log , p}(z)=\left( \left( \log \frac{e}{1-|z|^2}\right) ^{1-\frac{1}{p}}\right) ^{-1}, \end{aligned}$$

it is not difficult to see that $\widetilde{v}_{\log , p}=v_{\log , p}.$ Indeed, it is clear that $\widetilde{v}(z)=v(z),$ when (see [4])

$$\begin{aligned} v(z)=\left( \max \{|g(w)|; |w|=|z|\}\right) ^{-1} \end{aligned}$$

is a weight for some $g\in H(\mathbb {D}).$ Hence, the statement follows with $g(z)=(\log \frac{e}{1-|z|^2})^{1-\frac{1}{p}}.$

Lemma 3.1

([5]) For $\alpha >0$, we have $\lim _{k\rightarrow \infty }k^\alpha \Vert z^{k-1}\Vert _{v_\alpha }=(\frac{2\alpha }{e})^\alpha .$

After a calculation, we get the following result.

Lemma 3.2

For $1<p<\infty$, we have $\lim _{k\rightarrow \infty }(\log k)^{1-\frac{1}{p}}\Vert z^{k}\Vert _{v_{\log , p}} \thickapprox 1.$

Lemma 3.3

([12]) 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

  $$\begin{aligned} \sup _{z\in {\mathbb D}}\frac{w(z)}{\widetilde{v}(\varphi (z))}|u(z)|<\infty . \end{aligned}$$

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

  $$\begin{aligned} \Vert uC_\varphi \Vert _{e, H_{v}^{\infty }\rightarrow H_{w}^{\infty }}=\lim _{s\rightarrow 1}\sup _{|\varphi (z)|>s}\frac{w(z)}{\widetilde{v}(\varphi (z))}|u(z)|.\nonumber \end{aligned}$$

Lemma 3.4

([4]) 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

  $$\begin{aligned} \sup _{k\ge 0}\frac{\Vert u \varphi ^k\Vert _w}{\Vert z^k\Vert _v}<\infty , \end{aligned}$$

  with the norm comparable to the above supermum.

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

  $$\begin{aligned} \Vert uC_\varphi \Vert _{e,H_{v}^{\infty }\rightarrow H_{w}^{\infty } }=\limsup _{k\rightarrow \infty }\frac{\Vert u \varphi ^k\Vert _w}{\Vert z^k\Vert _v}.\nonumber \end{aligned}$$

Theorem 3.1

Let $1<p<\infty , u \in H(\mathbb {D})$ and $\varphi \in S({\mathbb D})$. Then, $uC_\varphi : B_p \rightarrow \mathcal {B}$ is bounded if and only if $u\in \mathcal {B}$,

$$\begin{aligned} \sup _{j\ge 1} \Vert I_u(\varphi ^j)\Vert _{\mathcal {B}}<\infty ~~~~~~~\text{ and }~~~~~~ \sup _{j\ge 1} (\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^j)\Vert _{\mathcal {B}} <\infty . \nonumber \end{aligned}$$

Here

$$\begin{aligned} I_u f(z)=\int _0^z f'(\zeta )u(\zeta )d\zeta , J_u f(z)=\int _0^z f(\zeta )u'(\zeta )d\zeta . \end{aligned}$$

Proof

By Theorem 2.1 of [2], $uC_\varphi : B_p \rightarrow \mathcal {B}$ is bounded if and only if

$$\begin{aligned}&\sup _{z\in \mathbb {D}}\frac{\left( 1-|z|^2\right) |u'(z)|}{\Big (\Big (\log \frac{e}{1-|\varphi (z)|^2}\Big )^{1-\frac{1}{p}}\Big )^{-1}}<\infty ~~~~~~~~\text{ and }~~~~ \sup _{z\in \mathbb {D}}\frac{\left( 1-|z|^2\right) |u(z)||\varphi '(z)|}{1-|\varphi (z)|^2}<\infty . \nonumber \\ \end{aligned}$$

(3.1)

By Lemma 3.3, the first inequality in (3.1) is equivalent to the operator $u'C_\varphi : H^\infty _{v_{\log , p}}\rightarrow H^\infty _{v_1}$ is bounded. By Lemma 3.4, this is equivalent to

$$\begin{aligned} \sup _{j\ge 1}\frac{\Vert u' \varphi ^{j-1}\Vert _{v_1}}{\Vert z^{j-1}\Vert _{v_{\log , p}}}<\infty . \end{aligned}$$

The second inequality in (3.1) is equivalent to the weighted composition operator $u\varphi 'C_\varphi : H^\infty _{v_1}\rightarrow H^\infty _{v_1}$ is bounded. By Lemma 3.4, this is equivalent to

$$\begin{aligned} \sup _{j\ge 1}\frac{\Vert u\varphi ' \varphi ^{j-1}\Vert _{v_1}}{\Vert z^{j-1}\Vert _{v_1}}<\infty . \end{aligned}$$

Since $I_ug(0)=0, J_ug(0)=0$,

$$\begin{aligned} \Big ( I_u(\varphi ^j)(z)\Big )'=ju(z)\varphi '(z) \varphi ^{j-1}(z), ~~~~ \Big ( J_u(\varphi ^{j-1})(z)\Big )'=u'(z) \varphi ^{j-1}(z), \end{aligned}$$

by Lemmas 3.1 and 3.2, we see that $uC_\varphi : B_p \rightarrow \mathcal {B}$ is bounded if and only if

$$\begin{aligned} \sup _{j\ge 1} \Vert I_u(\varphi ^j)\Vert _{\mathcal {B}}= \sup _{j\ge 1} j\Vert u\varphi ' \varphi ^{j-1}\Vert _{v_1} \approx \sup _{j\ge 1}\frac{j\Vert u\varphi ' \varphi ^{j-1}\Vert _{v_1}}{j\Vert z^{j-1}\Vert _{v_1}}<\infty \end{aligned}$$

and

$$\begin{aligned} \infty> & {} \sup _{j\ge 1}\frac{\Vert u' \varphi ^{j-1}\Vert _{v_1}}{\Vert z^{j-1}\Vert _{v_{\log , p}}}=\sup _{j\ge 1} \frac{\Vert J_u(\varphi ^{j-1})\Vert _{\mathcal {B}}}{\Vert z^{j-1}\Vert _{v_{\log , p}}}\\\approx & {} \max \Big \{ \Vert u\Vert _{\mathcal {B}}, ~~\, ~~\,~ \sup _{j\ge 2} (\log (j-1))^{1-\frac{1}{p}}\Vert J_u(\varphi ^{j-1})\Vert _{\mathcal {B}} \Big \}\\= & {} \max \Big \{ \Vert u\Vert _{\mathcal {B}}, ~~\, ~~\,~ \sup _{j\ge 1} (\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^j)\Vert _{\mathcal {B}} \Big \}. \end{aligned}$$

The proof is completed.$\square$

Theorem 3.2

Let $1< p< \infty , u \in H(\mathbb {D})$ and $\varphi \in S({\mathbb D})$ such that $uC_\varphi : B_p \rightarrow \mathcal {B}$ is bounded. Then

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, B_p \rightarrow \mathcal {B} } \approx \max \Big \{\limsup _{j\rightarrow \infty } \Vert I_u(\varphi ^j)\Vert _{\mathcal {B}},~~~\, ~~\,~~ \limsup _{j\rightarrow \infty } (\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^{j})\Vert _{\mathcal {B}}\Big \}. \nonumber \end{aligned}$$

Proof

From the proof of Theorem 3.1 we see that the boundedness of $uC_\varphi : B_p \rightarrow \mathcal {B}$ is equivalent to the boundedness of the operators $u\varphi 'C_\varphi : H^\infty _{v_1}\rightarrow H^\infty _{v_1}$ and $u'C_\varphi : H^\infty _{v_{\log , p}}\rightarrow H^\infty _{v_1}$. By Lemmas 3.1, 3.2 and 3.4, we get

$$\begin{aligned} \Vert u\varphi 'C_\varphi \Vert _{e, H^\infty _{v_1}\rightarrow H^\infty _{v_1}}= & {} \limsup _{j\rightarrow \infty }\frac{\Vert u\varphi ' \varphi ^{j-1}\Vert _{v_1}}{\Vert z^{j-1}\Vert _{v_1}}= \limsup _{j\rightarrow \infty }\frac{j\Vert u\varphi ' \varphi ^{j-1}\Vert _{v_1}}{j\Vert z^{j-1}\Vert _{v_1}} \nonumber \\\approx & {} \limsup _{j\rightarrow \infty }j\Vert u\varphi ' \varphi ^{j-1}\Vert _{v_1} = \limsup _{j\rightarrow \infty }\Vert I_u(\varphi ^j)\Vert _{\mathcal {B}} \end{aligned}$$

(3.2)

and

$$\begin{aligned} \Vert u'C_\varphi \Vert _{e, H^\infty _{v_{\log , p}}\rightarrow H^\infty _{v_1}}= & {} \limsup _{j\rightarrow \infty }\frac{\Vert u' \varphi ^{j-1}\Vert _{v_1}}{\Vert z^{j-1}\Vert _{v_{\log , p}}}=\limsup _{j\rightarrow \infty }\frac{(\log (j-1))^{1-\frac{1}{p}}\Vert u' \varphi ^{j-1}\Vert _{v_1}}{(\log (j-1))^{1-\frac{1}{p}}\Vert z^{j-1}\Vert _{v_{\log , p}}}\nonumber \\\approx & {} \limsup _{j\rightarrow \infty }(\log (j-1))^{1-\frac{1}{p}}\Vert u' \varphi ^{j-1}\Vert _{v_1}\nonumber \\= & {} \limsup _{j\rightarrow \infty }(\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^j)\Vert _{\mathcal {B}}. \end{aligned}$$

(3.3)

The upper estimate From the fact that

$$\begin{aligned} (uC_\varphi f)'(z)=u'(z)f(\varphi (z))+u(z)\varphi '(z)f'(\varphi (z)), \end{aligned}$$

it is easy to see that

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, B_p \rightarrow \mathcal {B} } \le \Vert u'C_\varphi \Vert _{e, H^\infty _{v_{\log , p}}\rightarrow H^\infty _{v_1}}+\Vert u\varphi 'C_\varphi \Vert _{e, H^\infty _{v_1}\rightarrow H^\infty _{v_1}}. \end{aligned}$$

(3.4)

Then, by (3.2), (3.3) and (3.4) we get

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, B_p \rightarrow \mathcal {B} }\lesssim & {} \limsup _{j\rightarrow \infty } \Vert I_u(\varphi ^j)\Vert _{\mathcal {B}}+ \limsup _{j\rightarrow \infty } (\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^j)\Vert _{\mathcal {B}}\nonumber \\\lesssim & {} \max \Big \{\limsup _{j\rightarrow \infty } \Vert I_u(\varphi ^j)\Vert _{\mathcal {B}}, ~~\, ~~\,~ \limsup _{j\rightarrow \infty } (\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^j)\Vert _{\mathcal {B}}\Big \}. \nonumber \end{aligned}$$

The lower estimate From Theorem 2.1, (3.2), (3.3) and Lemma 3.3, we have

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, B_p \rightarrow \mathcal {B}}\gtrsim & {} F = \Vert u\varphi 'C_\varphi \Vert _{e, H^\infty _{v_1}\rightarrow H^\infty _{v_1}} \approx \limsup _{j\rightarrow \infty }\Vert I_u(\varphi ^j)\Vert _{\mathcal {B}} \nonumber \end{aligned}$$

and

$$\begin{aligned}&\Vert uC_\varphi \Vert _{e, B_p \rightarrow \mathcal {B} } \gtrsim E =\Vert u'C_\varphi \Vert _{e, H^\infty _{v_{\log , p}}\rightarrow H^\infty _{v_1}} \approx \limsup _{j\rightarrow \infty } (\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^{j})\Vert _{\mathcal {B}}. \nonumber \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert uC_\varphi \Vert _{e, B_p \rightarrow \mathcal {B}} \gtrsim \max \Big \{\limsup _{j\rightarrow \infty } \Vert I_u(\varphi ^j)\Vert _{\mathcal {B}},~~\, ~~\,~ \limsup _{j\rightarrow \infty } (\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^{j})\Vert _{\mathcal {B}}\Big \}. \nonumber \end{aligned}$$

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

From Theorem 3.2, we immediately get the following result.

Theorem 3.3

Let $1<p<\infty , u \in H(\mathbb {D})$ and $\varphi \in S({\mathbb D})$ such that $uC_\varphi : B_p \rightarrow \mathcal {B}$ is bounded. Then $uC_\varphi : B_p \rightarrow \mathcal {B}$ is compact if and only if

$$\begin{aligned} \limsup _{j\rightarrow \infty } \Vert I_u(\varphi ^j)\Vert _{\mathcal {B}}=0~~~\text{ and }~~~ \limsup _{j\rightarrow \infty } (\log j)^{1-\frac{1}{p}}\Vert J_u(\varphi ^{j})\Vert _{\mathcal {B}} =0. \nonumber \end{aligned}$$