Volterra integral operator and essential norm on Dirichlet type spaces

1.   Introduction

First, let us introduce some necessary notation. Let ${\mathbb{D}}$ be the unit disk in the complex plane ${\mathbb{C}}$, $H({\mathbb{D}})$ be the class of functions analytic in ${\mathbb{D}}$ and $H^\infty$ be the class of bounded analytic functions on ${\mathbb{D}}$. The Bloch space ${\mathcal{B}}$ (^[34]) is the class of all $f\in H({\mathbb{D}})$ for which

The little Bloch space ${\mathcal{B}}_0,$ consists of all $f\in{\mathcal H}({\mathbb{D}})$ such that

The Hardy space $H^p({\mathbb{D}})$ $(0 < p < \infty)$ (^[8,10]) is the set of $f\in H({\mathbb{D}})$ with

Suppose that $0 < p < \infty$, $\alpha > -1$ and $dA_{\alpha}(z) = (1-|z|^2)^\alpha dA(z) = \frac{1}{\pi}(1-|z|^2)^\alpha dxdy$. The weighted Bergman space $A^p_{\alpha}({\mathbb{D}})$ (^[34]) is the set of $f\in H({\mathbb{D}})$ with

Let $\alpha\geq 0$. The Dirichlet type space $\mathcal{D}_{\alpha}$ is the set of $f\in H({\mathbb{D}})$ with

If $\alpha = 0$, it gives classic Dirichlet space $\mathcal{D}$. When $\alpha = 1$, it is Hardy space $H^2$. When $\alpha > 1$, it turns into weighted Bergman spaces $A_{\alpha-2}^2$. Thus, the interesting scope is $\alpha\in(0, 1)$. For more information relating to $\mathcal{D}_{\alpha}$, we refer to ^[23,25,26].

In this paper, we use the weighted function in ^[9,30]. We always suppose that $K:[0, \infty)\rightarrow [0, \infty)$ is a right-continuous and nondecreasing function. The weighted function $K$ also satisfies

and

where

Let $\alpha\geq 0$ and Dirichlet type space ${\mathcal{D}_{K, \alpha}}$ denotes the spaces of function $f\in H({\mathbb{D}})$ satisfying

When $\alpha > 0$, if the weighted function $K$ satisfies $(A)$ and $(B)$, we easily to see that $\mathcal{D}_\alpha\subseteq{\mathcal{D}_{K, \alpha}}\subseteq A^2_{\alpha-1}$. By ^[9], there exist a small $c > 0$, such that $C_1t^{1-c}\leq K(t)\leq C_2 t^{c}$, where $0 < t < 1$, $C_1 > 0$ and $C_2 > 0$. Thus, when $\alpha\geq 1$, we easily to see that $A^2_{\alpha-2+c}\subseteq{\mathcal{D}_{K, \alpha}}\subseteq A^2_{\alpha-1-c}$. Moreover, using high order characterization, it is not hard to check that ${\mathcal{D}_{K, \alpha}}$ turns into a Bergman type space, when $\alpha\geq 1$. Thus, the interesting scope is $\alpha\in[0, 1)$. For more results of ${\mathcal{D}_{K, \alpha}}$ spaces, we refer to ^{[3,4,5,11,15,19,20]}.

Let $I$ be an arc of ${\partial {\mathbb{D}}}$ and $|I|$ be the normalized Lebesgue arc length of $I.$ The Carleson square based on $I,$ denoted by $S(I),$ is defined by

Let $\mu$ be a positive Borel measure on ${\mathbb{D}}.$ For $0 < s < \infty,$ $\mu$ is called an $s$-Carleson measure if

We say that a function $f\in H^2({\mathbb{D}})$ belongs to Morrey type space $\mathcal{H}_K^2$ if

where

This space was introduced by H. Wulan and J. Zhou in ^[29]. When $K(t) = t$, it gives the $BMOA$ space, the space of those analytic functions $f$ in the Hardy space $H^p$ whose boundary functions having bounded mean oscillation on $\partial{\mathbb{D}}$. In the case $K(t) = t^{\lambda}$, $0 < \lambda < 1$, the space ${\mathcal H}_K^2$ gives classical Morrey space $\mathcal{L}^{2, \lambda}$. Morrey spaces $\mathcal{L}^{2, \lambda}$ were introduced by Morrey in ^[16]. From ^[29], we know that $f\in \mathcal{H}_K^2$ if and only if

where $\varphi_a(z) = \frac{a-z}{1-\overline{a}z}$.

Let $\alpha\geq 0$ and we say that a function $f\in H({\mathbb{D}})$ belongs to Morrey type space ${\mathcal{H}_{K, \alpha}^2}$ if

It is easy to verify ${\mathcal{H}_{K, \alpha}^2}$ is a Banach space under the above norm.

Let $f, g\in H({\mathbb{D}})$. The Volterra integral operator $V_g$ and the integral operator $S_g$ are defined by

respectively. For $g\in H({\mathbb{D}})$, the multiplication operator $M_g$ is defined by $M_gf(z) = f(z)g(z)$. It is easy to see that $M_g$ is related with $S_g$ and $V_g$ by

It it well known that $V_g$ is bounded on the Hardy space $H^p$ (Bergman space $A^p$) if and only if $g\in BMOA$ ($g\in \mathcal{B}$). $V_g$ is bounded on $BMOA$ if and only if $g\in BMOA_{\log}$ (see ^[24]). For more information relating to Volterra integral operator $V_g$, we refer to ^{[1,2,7,12,13,14,17,21,22,28,31,33]}.

In this note, we study Volterra integral operator $V_g$ acting from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$, that is, we prove that $V_g: {\mathcal{D}_{K, \alpha}}\rightarrow {\mathcal{H}_{K, \alpha}^2}$ is bounded if and only if $g\in {\mathcal{B}}$, when $0 < \alpha < 1$. Meanwhile, the boundedness of $S_g$ and the essential norm of $V_g$ and $S_g$ from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$ are also studied.

In this paper, the symbol $f\approx g$ means that $f\lesssim g\lesssim f$. We say that $f\lesssim g$ if there exists a constant $C$ such that $f\leq Cg$.

2.   Auxiliary results

In this section, we are going to give some auxiliary results.

Lemma 1. Let (A) and (B) hold for $K$. Suppose that $\alpha > 0$ and $f\in {\mathcal{D}_{K, \alpha}}$, then

Proof. The proof is similar to ^[33], thus we omit it here. The proof is completed.

Lemma 2. Let $(B)$ hold for $K$. Suppose that $\alpha > 0$. Then

and

where $z, \ a\in{\mathbb{D}}.$

Proof. Since $(B)$ holds, then from ^[9], there is some $c\in (0, 1)$, such that

Combining with $K$ which is nondecreasing and Lemma 3.10 of ^[34], we obtain

where the third inequality is deduced by $1-|a|\leq |1-\overline{a}z|$ and $K$ is nondecreasing, the last second inequality is deduced by $1-|z|\leq |1-\overline{a}z|$ and $(1)$. Thus, $f_a\in {\mathcal{D}_{K, \alpha}}$. Similar proof can be applied to $F_a$, thus we omit here. The proof is completed.

Lemma 3. (^[34]) Suppose that $\alpha > -1$ and $\mu$ is a non-negative measure on ${\mathbb{D}}$. Then $\mu$ is a $(2+\alpha)$-Carleson measure if and only if the following inequality

holds for all $f\in A_{\alpha}^2$.

Lemma 4. (^[32]) Let $p > 1$ and $f\in H({\mathbb{D}})$. Then $f\in {\mathcal{B}}$ if and only if the measure $d\mu_f = |f'(z)|^2(1-|z|^2)^pdA(z)$ is a $p$-Carleson measure.

Lemma 5. (^[6]) Suppose that $1 < p < \infty$, $\alpha > -1$, $\beta\geq 0$ with $\beta < 2+\alpha$. Let $f\in H({\mathbb{D}})$ and $z, w\in{\mathbb{D}}$. Then

Lemma 6. Let (A) and (B) hold for $K$. Suppose that $0 < \alpha < 1$. Then $f\in{\mathcal{H}_{K, \alpha}^2}$ if and only if

Proof. The proof is similar to Lemma 2.1 of ^[18]. Thus we omit here. The proof is complete.

3.   Boundedness of $V_g$ and $S_g$ operators

Theorem 1. Let (A) and (B) hold for $K$. Suppose that $g\in H({\mathbb{D}})$ and $0 < \alpha < 1$. Then $V_g$ is bounded from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$ if and only if $g\in {\mathcal{B}}$. Moreover, the operator norm satisfies $\|V_g\|\approx\|g\|_{{\mathcal{B}}}$.

Proof. For any $I\in{\partial {\mathbb{D}}}$, let $a = (1-|I|)\zeta\in{\mathbb{D}}$, where $\zeta$ is the center of $I$. Then

Let $f_a$ be defined as in Lemma 2. Then

Suppose that $V_g$ is bounded from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$. By Lemmas 4 and 6, we have

Thus, $g\in {\mathcal{B}}$.

On the other hand, suppose that $g\in {\mathcal{B}}$, by Lemma 4, we have $d\mu_g = |g'(z)|^2(1-|z|^2)^{\alpha+1}dA(z)$ is a $(\alpha+1)$-Carleson measure. Let $f\in {\mathcal{D}_{K, \alpha}}$. From Lemma 6, we only need to prove that

Since

Using Lemma 1 and $(3)$, we see that

By Lemma 3, we have $A_{\alpha-1}^2\subseteq L^2(d\mu_g)$. Note that

Thus, ${\mathcal{D}_{K, \alpha}}\subseteq A_{\alpha-1}^2$. Bearing in mind these facts, we are going to estimate $N$. Let $z = \varphi_a(w)$. Since $|\varphi_a'(w)|(1-|w|^2) = 1-|\varphi_a(w)|^2,$ using Lemmas 3, 4, 5, we obtain

where the last second inequality is deduced by $(1)$. Combining the estimates $M$ and $N,$ we conclude that $V_g:{\mathcal{D}_{K, \alpha}}\to {\mathcal{H}_{K, \alpha}^2}$ is bounded.

Theorem 2. Let (A) and (B) hold for $K$. Suppose that $g\in H({\mathbb{D}})$ and $0 < \alpha < 1$. Then $S_g$ is bounded from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$ if and only if $g\in H^{\infty}$. Moreover, the operator norm satisfies $\|S_g\|\approx\sup_{z\in{\mathbb{D}}}|g(z)|$.

Proof. Suppose that $S_g$ is bounded from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$. Let $a\in{\mathbb{D}}$ and

By Lemma 2, we have $F_a\in{\mathcal{D}_{K, \alpha}}$ and $\|F_a\|_{{\mathcal{D}_{K, \alpha}}}\lesssim 1$. For $a\in {\mathbb{D}}$ and $r > 0$, let $D(a, r)$ denote the Bergman metric disk centered at $a$ with radius $r$. From ^[34] we see that

when $z\in D(a, r)$. Using subharmonic property of $|g|^2$, we have

That is,

Since $a\in{\mathbb{D}}$ is arbitrary, we have

On the other hand. Let $g\in H^{\infty}$. Using (1), we can deduce that for $f\in{\mathcal{D}_{K, \alpha}}$,

The proof is completed.

Remark. Note that

Hence, if (A) and (B) hold for $K$, then $M_g$ is bounded from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$ if and only if $g\in H^{\infty}$.

4.   Essential norm

Let us recall the definition of essential norm. Suppose that $X$ be a Banach space and $T$ is a bounded linear operator on $X$. The essential norm of $T$ is the distance of $T$ to the closed ideals of compact operators, that is

Note that $T$ is compact if and only if $\|T\|_{e} = 0$.

Lemma 7. Suppose that $0 < \alpha < 1$ and $K$ satisfies the conditions (A) and (B). Let $g\in {\mathcal{B}}$. Then $V_{g_r}:{\mathcal{D}_{K, \alpha}}\rightarrow {\mathcal{H}_{K, \alpha}^2}$ is compact. Here $g_r(z) = g(rz),$ $0 < r < 1, z\in{\mathbb{D}}.$

Proof. Let $\{f_n\}$ be any function sequence such that $\|f_n\|_{{\mathcal{D}_{K, \alpha}}}\lesssim1$ and $f_n\rightarrow0$ uniformly on compact subsets of ${\mathbb{D}}$ as $n\rightarrow \infty$. We need only to show that

Since

Combining with (1), we have

Note that $\|f_n\|_{{\mathcal{D}_{K, \alpha}}}\lesssim1$ and Lemma 1, the argument is then finished by the Dominated Convergence Theorem.

Let $X$ and $Y$ be two Banach spaces with $X\subset Y$. If $f\in Y$, then the distance from $f$ to $X$ is defined as

We also need the following lemma.

Lemma 8. (^[27]) If $f\in{\mathcal{B}},$ then

Theorem 3. Suppose $0 < \alpha < 1$, $g\in{\mathcal{B}}$ and $K$ satisfy the conditions (A) and (B). Then $V_g : {\mathcal{D}_{K, \alpha}}\rightarrow {\mathcal{H}_{K, \alpha}^2}$ satisfies

Proof. Let $\{I_n\}$ be the subarc sequence of ${\partial {\mathbb{D}}}$, such that $|I_n|\rightarrow0$ as $n\rightarrow \infty$, $w_n = (1-|I_n|)\zeta_n\in{\mathbb{D}}$, where $\zeta_n$ is the center of $I_n$. $n = 1, 2, ....$ Then

Thus, by ^[9], we know that

Take

Then $f_{n}\rightarrow0$ uniformly on the compact subsets of ${\mathbb{D}}$ as $n\rightarrow \infty$ and $\|f_{n}\|_{{\mathcal{D}_{K, \alpha}}}\lesssim1$. Thus, for any compact operator $S$ from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$, we have

Therefore

On the other hand, by Lemma 7, $V_{g_r}: {\mathcal{D}_{K, \alpha}}\rightarrow {\mathcal{H}_{K, \alpha}^2}$ is compact operator. Combining this with Theorem 1 and the linearity of $V_g$ respect to $g$ implies

Together with Lemma 8, we have

The proof is completed.

Corollary 1. Suppose $0 < \alpha < 1$ and $K$ satisfies the conditions (A) and (B). If $g\in H({\mathbb{D}}),$ then $V_g:{\mathcal{D}_{K, \alpha}}\to{\mathcal{H}_{K, \alpha}^2}$ is compact if and only if $g\in{\mathcal{B}}_0.$

Theorem 4. Suppose $0 < \alpha < 1$ and $K$ satisfies the conditions (A) and (B). If $g\in H({\mathbb{D}})$ and $S_g$ is bounded from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$, then

Proof. For compact operators $S$, it follows that

On the other hand, we choose the sequence $\{a_n\}\subset {\mathbb{D}}$ such that $|a_n|\rightarrow 1$. We define

It follows from the proof of Lemma 2 that $\|f_{n}\|_{{\mathcal{D}_{K, \alpha}}}\lesssim1$. It is easy to check that $f_{n}$ converges to zero uniformly on any compact subsets of ${\mathbb{D}}$. Then $\|Sf_{n}\|_{{\mathcal{H}_{K, \alpha}^2}}\rightarrow0$ as $n\rightarrow \infty$ for any compact operator $S$ from ${\mathcal{D}_{K, \alpha}}$ to ${\mathcal{H}_{K, \alpha}^2}$. So

From the proof of Theorem 2, we have

Since $\{a_n\}\subseteq{\mathbb{D}}$ is arbitrary, we have

The proof is completed.

Corollary 2. Suppose $0 < \alpha < 1$ and $K$ satisfy the conditions (A) and (B). If $g\in H({\mathbb{D}}),$ then $S_g:{\mathcal{D}_{K, \alpha}}\to{\mathcal{H}_{K, \alpha}^2}$ is compact if and only if $g = 0.$

5.   Conclusions

In this paper, we give some equivalent characterizations of Volterra integral operator and essential norm from Dirichlet type spaces ${\mathcal{D}_{K, \alpha}}$ to Morrey type spaces ${\mathcal{H}_{K, \alpha}^2}$.

Acknowledgments

The authors thank the referee for useful remarks and comments that led to the improvement of this paper. This work was supported by NNSF of China (No. 11801250, No.11871257), Overseas Scholarship Program for Elite Young and Middle-aged Teachers of Lingnan Normal University, Yanling Youqing Program of Lingnan Normal University (No. YL20200202), the Key Program of Lingnan Normal University (No. LZ1905), The Innovation and developing School Project of Guangdong Province (No. 2019KZDXM032) and Education Department of Shaanxi Provincial Government (No.19JK0213).

Conflict of interest

We declare that we have no conflict of interest.