Embedding of Q_p spaces into tent spaces and Volterra integral operator

1.   Introduction

Let ${\mathbb{D}}$ be the open unit disk in the complex plane ${\mathbb{C}}$ and $H({\mathbb{D}})$ be the class of all functions analytic in ${\mathbb{D}}$. Let $0 < p < \infty$ and $-1 < \alpha < \infty$. The Dirichlet type space ${\mathcal{D}}_{\alpha}^p$ is the set of all $f\in H({\mathbb{D}})$ such that

where $dA$ is the normalized Lebesgue area measure in ${\mathbb{D}}$ such that $A({\mathbb{D}}) = 1$. When $p = 2$ and $\alpha = 0$, it gives the classic Dirichlet space $\mathcal{D}$. When $p = 2$ and $\alpha = 1$, it gives the Hardy space $H^2$. When $\alpha = p$, ${\mathcal{D}}_{\alpha}^p$ is just the classical Bergman space $A^p$.

Let $0 < p < \infty$. The ${Q_p}$ space is the space of all $f\in H({\mathbb{D}})$ such that (see, e.g., ^[23])

where $\sigma_a(z) = \frac{a-z}{1-\overline{a}z}$. When $p > 1$, ${Q_p}$ is the Bloch space $\mathcal{B}$ (see ^[24,25]), which denote the space of all $f\in H({\mathbb{D}})$ such that

When $p = 1$, $Q_1 = BMOA$, the space of analytic functions in the Hardy space $H^1({\mathbb{D}})$ whose boundary functions have bounded mean oscillation (see, e.g., ^[25]).

Let $0 < p, s < \infty$, $-2 < q < \infty$. A function $f \in H({\mathbb{D}})$ is said to belong to $\mathcal{F}(p, q, s)$ if

An $f \in \mathcal{F}_0(p, q, s)$ if $f \in H({\mathbb{D}})$ and

$\mathcal{F}(p, q, s)$ is a Banach space under the norm $\|\cdot\|_{\mathcal{F}(p, q, s)}$ when $p\geq1$. This space was first introduced by Zhao in ^[24] and called general function space because it can get many function spaces if it takes special parameters of $p, q, s$. From ^[24] we see that $\mathcal{F}(p, p-2, s)$ is just the Bloch space when $s > 1$.

For $0 < q, s < \infty$, let $\mathcal{LF}(q, q-2, s)$ denote the space of all $f\in H({\mathbb{D}})$ such that

It is easy to check that $\mathcal{LF}(q, q-2, s)$ is a Banach space under the norm $\|\cdot\|_{\mathcal{LF}(q, q-2, s)}$ when $q\geq1$.

Let $g\in H({\mathbb{D}}).$ The Volterra integral operator $T_g$, which introduced by Pommerenke in ^[13], was defined by

The importance of the operator $T_g$ comes from the fact that $T_g f+I_g f = M_g f-f(0)g(0),$ where the operators $M_g$ and $I_g$ are defined by

respectively. Note that the integral form of the classical Cesàro operator $\mathcal{C}$ is

Hence the operator $T_g$ can also be seen as the generalization of the Cesàro operator $\mathcal{C}$. In ^[13], Pommerenke showed that $T_g$ is bounded on $H^2$ if and only if $g\in BMOA$. In ^[2], Aleman and Siskakis showed that $T_g$ is bounded (compact) on $A^p$ if and only if $g\in \mathcal{B}\, (g\in \mathcal{B}_0)$. Recently, the operator $T_g$ has been received many attention. See ^{[1,2,4,5,6,7,8,12,14,15,18,19,22,24]} and the references therein for more study of the operator $T_g$.

For an arc $I \subseteq \partial \mathbb{D}$, let $|I| = \frac{1}{2\pi}\int_{I}|d\zeta|$ be the normalized length of $I$. Let $0 < \alpha < \infty$ and $\mu$ be a positive Borel measure on ${\mathbb{D}}$. As usual, we say that $\mu$ is a $\alpha$-Carleson measure if

where $S(I) = \{z\in {\mathbb{D}}: 1-|I|\leq |z| < 1, \ \ \ \frac{z}{|z|}\in I\}$ is the Carleson box based on $I$. When $\alpha = 1$, it gives the classical Carleson measure. $\mu$ is said to be a vanishing $\alpha$-Carleson measure if $\lim_{|I|\rightarrow 0}\frac{\mu(S(I))}{|I|^{\alpha}} = 0.$

Let $0 < \lambda, q < \infty$, $s\geq 0$ and $\mu$ be a positive Borel measure on ${\mathbb{D}}$. The tent space ${\mathcal{T}}_{\lambda, s}^{q}(\mu)$ consists of all $f\in H({\mathbb{D}})$ satisfied

The tent space ${\mathcal{T}}_{\lambda, s}^{q}(\mu)$ was introduced by Liu, Lou and Zhu in ^[10]. When $q = 2$ and $s = 0$, ${\mathcal{T}}_{\lambda, 0}^{2}(\mu) = \mathcal{T}_\lambda^{\infty}$ was first introduced by Xiao in ^[22].

In ^[22], Xiao studied the inclusion mapping $i: Q_p\rightarrow \mathcal{T}_p^\infty(\mu)$. He showed that the inclusion mapping $i: Q_p\rightarrow \mathcal{T}_p^\infty(\mu)$ is bounded (resp. compact) if and only if

As an application, he proved that the operator $T_g:Q_p\rightarrow Q_p$ is bounded if and only if

In ^[10], Liu, Lou and Zhu studied the embedding of some Möbius invariant spaces, such as the Bloch space and the $Q_p$ space, into $\mathcal{T}_{\lambda, s}^{2}$. Among others, they proved the following Theorem A. See ^{[6,9,12,14,15,16,17,21]} and the references therein for more study of analytic function spaces embedding into various tent spaces.

Theorem A. Let $0 < p < 1$ and $\mu$ be a positive Borel measure on ${\mathbb{D}}$. If ${Q_p}$ is continuously contained in $\mathcal{T}_{p, 2}^2$, then $\mu$ is a $p$-Carleson measure. If ${\mathcal{D}}^2_p$ is continuously contained in $L^2({\mathbb{D}}, d\mu)$, then ${Q_p}$ is continuously contained in $\mathcal{T}_{p, 2}^2$.

By [22,Lemma 2.1 (ⅱ)], we see that $\mu$ is a $p$-Carleson measure if ${\mathcal{D}}^2_p$ is continuously contained in $L^2({\mathbb{D}}, d\mu)$. But the converse is not clear. The nature question then arise, what can one say if we change $\mathcal{T}_{p, 2}^{2}$ into $\mathcal{T}_{\lambda, s}^{q}$ when $q > 2$?

In this paper, we give an answer by using a new method, which was different to ^[10,22]. We study the boundedness and compactness of the inclusion mapping from ${Q_p}$ spaces into tent spaces $\mathcal{T}_{\frac{qp}{2}, s}^{q}$. As an application, we study the boundedness of Volterra integral operator $T_g$ acting from ${Q_p}$ to ${\mathcal{LF}(q, q-2, \frac{qp}{2})}$. Meanwhile, the compactness and essential norm of the operator $T_g$ acting from ${Q_p}$ to ${\mathcal{LF}(q, q-2, \frac{qp}{2})}$ are also investigated.

Throughout this paper, we say that $A\lesssim B$ if there exists a constant $C$ such that $A\leq CB$. The symbol $A\approx B$ means that $A\lesssim B\lesssim A$.

2.   Embedding of ${Q_p}$ spaces into tent spaces

In this section, we study the embedding from ${Q_p}$ to tent spaces. We give a complete characterization for the boundedness and compactness of the inclusion mapping $i:{Q_p}\rightarrow T_{\lambda, s}^q(\mu).$ We say that the inclusion mapping $i: {Q_p}\rightarrow \mathcal{T}_{\lambda, s}^q(\mu)$ is compact if

whenever $I\subseteq \partial{\mathbb{D}}$ and $\{f_n\}$ is a bounded sequence in ${Q_p}$ that converges to $0$ uniformly on compact subsets of ${\mathbb{D}}$.

The following result is one of the main results in this paper.

Theorem 1. Let $0 < p < 1$ and $\mu$ be a positive Borel measure. If $2 < q < \infty$ and $0 < s\leq q < \infty$, then the following statements hold.

(i) The inclusion mapping $i: \ {Q_p}\rightarrow \mathcal{T}_{\frac{qp}{2}, s}^q(\mu)$ is bounded if and only if

(ii) The inclusion mapping $i: \ {Q_p}\rightarrow \mathcal{T}_{\frac{qp}{2}, s}^q(\mu)$ is compact if and only if

Proof. $(i)$ Assume that the inclusion mapping $i: \ {Q_p}\rightarrow \mathcal{T}_{\frac{qp}{2}, s}^q(\mu)$ is bounded. For any fixed arc $I\subseteq {\partial {\mathbb{D}}}$, let $e^{i\theta}$ be the center of $I$ and $a = (1-|I|)e^{i\theta}.$ Set $f_a(z) = \log\frac{2}{(1-\overline{a}z)}$. Then $f_a\in{Q_p}$ and

whenever $z\in S(I).$ By the boundedness of $i$, we have

which implies (1), as desired.

Conversely, assume that (1) holds. Let $f\in {Q_p}$. For any fixed arc $I\subseteq {\partial {\mathbb{D}}}$, let $e^{i\theta}$ be the center of $I$ and $a = (1-|I|)e^{i\theta}.$ We have

where

Since $f\in {Q_p} \subseteq {\mathcal{B}}$, we obtain

which implies that for any $I\subseteq {\partial {\mathbb{D}}}$,

Since $0 < s\leq q < \infty$, we get

which implies that ${\mathcal{D}}^2_p\subseteq L^q(d\mu)$ by [4,Theorem 1]. Therefore,

By the growth of functions in ${Q_p}$ and

we deduce that

Thus, we only need to prove that

Since

we obtain $E\lesssim E_1+E_2$, where

and

Noting that

we have

By ^[11], we deduce that

Therefore, $E \lesssim\|f\|_{{Q_p}}^2$, as desired.

$(ii)$ Suppose that the inclusion mapping $i: \ Q_p\rightarrow \mathcal{T}_{\frac{qp}{2}, s}^q(\mu)$ is compact. Let $\{I_n\}\subseteq {\partial {\mathbb{D}}}$ and $|I_n|\rightarrow 0$ as $n\rightarrow\infty$. Suppose $e^{i\theta_n}$ is the center of $I_n$ and $a_n = (1-|I_n|)e^{i\theta_n}.$ Set $f_{a_n}(z) = \log\frac{2}{(1-\overline{a_n}z)}.$ Then $f_{a_n} \in{Q_p}$ and $\log\frac{2}{(1-\overline{a_n}z)} \approx\log\frac{2}{|I_n|}$. Therefore

which implies that (2) holds.

Conversely, assume that (2) holds. Then it is clear that

Let $\{f_n\}$ be a bounded sequence in ${Q_p}$ such that $\{f_n\}$ converges to zero uniformly on each compact subset of ${\mathbb{D}}.$ From ^[12] we have

as $r\to 1^{-}$ and $n\to\infty.$ Therefore, $\lim_{n\to\infty}\|f_n\|_{\mathcal{T}_{\frac{qp}{2}, s}^q(\mu)} = 0.$ This shows that the inclusion mapping $i:{Q_p}\to \mathcal{T}_{\frac{qp}{2}, s}^q(\mu)$ is compact.

In particular, let $s = q$, we get the following result.

Corollary 1. Let $0 < p < 1$, $2 < q < \infty$ and $\mu$ be a positive Borel measure. Then the inclusion mapping $i: \ {Q_p}\rightarrow \mathcal{T}_{\frac{qp}{2}, q}^q(\mu)$ is bounded (resp., compact) if and only if

3.   Volterra integral operator $T_g: \ {Q_p}\rightarrow{\mathcal{LF}(q, q-2, \frac{qp}{2})}$

In this section, we study the boundednss, compactness and the essential norm of Volterra integral operator $T_g: \ {Q_p}\rightarrow{\mathcal{LF}(q, q-2, \frac{qp}{2})}$. We need the following equivalent characterization of functions in $\mathcal{LF}(q, q-2, s)$.

Proposition 1. Let $1 < q < \infty$ and $0 < s < \infty$. Then $f\in \mathcal{LF}(q, q-2, s)$ if and only if

Proof. Let $f\in \mathcal{LF}(q, q-2, s)$. For any $I\in{\partial {\mathbb{D}}}$, let $a = (1-|I|)\zeta\in{\mathbb{D}}$, where $\zeta$ is the center of $I$. Then

Combining with $1-|\sigma_a(z)|^2 = \frac{(1-|a|^2)(1-|z|^2)}{|1-\overline{a}z|^2},$ we have

as desired.

Conversely, assume that (3) holds. For any given nonzero $a\in{\mathbb{D}}$, let $I_a$ be the subarc of ${\partial {\mathbb{D}}}$ with midpoint $a/|a|$ and length $1-|a|$; and for $a = 0$, let $I_a = {\partial {\mathbb{D}}}$. Moreover, let $J_n = 2^nI_a$ for $n = 0, 1, \cdots, N-1$, where $N$ is the smallest positive integer such that $2^N|I_a|\geq 1$. Then we have the following estimate:

and

Without loss of generality, we may assume $|a| > 3/4$. By (4) and (5) we have

The proof is complete.

Theorem 2. Let $0 < p < 1$, $2 < q < \infty$ and $g\in H({\mathbb{D}})$. Then $T_g: \ {Q_p}\rightarrow{\mathcal{LF}(q, q-2, \frac{qp}{2})}$ is bounded if and only if $g\in \mathcal{F}(q, q-2, \frac{qp}{2})$.

Proof. Suppose that $g\in \mathcal{F}(q, q-2, \frac{qp}{2})$. By ^[24] we have

which means that $d\mu_g(z) = |g'(z)|^{q}(1-|z|^2)^{q-2+\frac{qp}{2}}dA(z)$ is a $\frac{qp}{2}$-Carleson measure. Let $f\in{Q_p}$. By Corollary 1, we see that $i: \ {Q_p}\rightarrow \mathcal{T}_{\frac{qp}{2}, q}^q(\mu_g)$ is bounded, i.e.,

which together with Proposition 1 imply that

Therefore $T_g: \ {Q_p}\rightarrow{\mathcal{LF}(q, q-2, \frac{qp}{2})}$ is bounded.

Conversely, assume that $T_g: \ {Q_p}\rightarrow{\mathcal{LF}(q, q-2, \frac{qp}{2})}$ is bounded. For any fixed arc $I\subseteq {\partial {\mathbb{D}}}$ and let $e^{i\theta}$ be the center of $I$ and $a = (1-|I|)e^{i\theta}.$ Set $f_a(z) = \log\frac{2}{(1-\overline{a}z)}$. Then $f_a \in{Q_p}$ for $0 < p < \infty$. Since

when $z\in S(I)$, we get

which implies that $g\in \mathcal{F}(q, q-2, \frac{qp}{2})$ by ^[24]. The proof is complete.

Next, we give an estimation for the essential norm of $T_g$. First, we recall some definitions. Let $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ be Banach spaces and $T:X\to Y$ be a bounded linear operator. The essential norm of $T:X\to Y,$ denoted by $\|T\|_{e, X\to Y},$ is defined by

It is easy to see that $T:X\to Y$ is compact if and only if $\|T\|_{e, X\to Y} = 0$. Let $A$ be a closed subspace of $X.$ Given $f\in X,$ the distance from $f$ to $A$, denoted by ${\rm dist}_X(f, A),$ is defined by ${\rm dist}_X(f, A) = \inf_{g\in A}\|f-g\|_{X}.$

Lemma 1. Let $2 < q < \infty$ and $0 < \lambda < \infty$. If $g\in {\mathcal{F}(q, q-2, \lambda)}$, then

Here $g_r(z) = g(rz),$ $0 < r < 1, z\in{\mathbb{D}}.$

Proof. For any given $g\in {\mathcal{F}(q, q-2, \lambda)}$, then $g_r\in {\mathcal{F}_0(q, q-2, \lambda)}$ and $\|g_r\|_{{\mathcal{F}(q, q-2, \lambda)} }\lesssim\|g\|_{{\mathcal{F}(q, q-2, \lambda)} }.$ Let $\delta\in (0, 1)$. We choose $a\in (0, \delta).$ Then $\sigma_a(z)$ lies in a compact subset of ${\mathbb{D}}$. So

Making a change of variables, we have

By the definition of distance, we obtain

Denote by $\psi_{r, a}(z) = \sigma_{ra}\circ r \sigma_a(z)$. Then $\psi_{r, a}$ is an analytic self-map of ${\mathbb{D}}$ and $\psi_{r, a}(0) = 0$. Making a change variable of $z = \sigma_a(z)$ and applying the Littlewood's subordination theorem (see Theorem 1.7 of ^[3]), we have

Since $\delta$ is arbitrary, we get

On the other hand, for any $g\in {\mathcal{F}(q, q-2, \lambda)}$,

which, together with (3.4), implies the desired result. The proof is complete.

Lemma 2. Let $0 < p < 1$ and $2 < q < \infty$. If $0 < r < 1$ and $g\in \mathcal{F}(q, q-2, \frac{qp}{2}),$ then $T_{g_r}:{Q_p}\to{\mathcal{LF}(q, q-2, \frac{qp}{2})}$ is compact.

Proof. Given $\{f_n\}\subset{Q_p}$ such that $\{f_n\}$ converges to zero uniformly on any compact subset of ${\mathbb{D}}$ and $\sup_{n}\|f_n\|_{{Q_p}}\leq 1.$ Then by the following well-known inequality

we get

By the dominated convergence theorem, we get the desire result. The proof is complete.

The following result is an important tool to study the essential norm and compactness of operators on some analytic function spaces, see ^[20].

Lemma 3. 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 for any bounded sequence $\{f_n\}$ in $X$ such that $\{f_n\}$ converges to zero uniformly on every compact set of ${\mathbb{D}}$, then the sequence $\{Tf_n\}$ converges to zero in the norm of $Y$.

Theorem 3. Let $0 < p < 1$, $2 < q < \infty$ and $g\in H({\mathbb{D}})$. If $T_g: \ {Q_p}\rightarrow{\mathcal{LF}(q, q-2, \frac{qp}{2})}$ is bounded, then

Proof. Let $\{I_n\}\subseteq {\partial {\mathbb{D}}}$ and $|I_n|\rightarrow 0$ as $n\rightarrow\infty$. Suppose $e^{i\theta_n}$ is the center of $I_n$ and $w_n = (1-|I_n|)e^{i\theta_n}.$ For each $n,$ let

Then $|f_{w_n}(z)|\approx\log\frac{2}{|I_n|}$ when $z\in S(I_n)$ and $\{f_{w_n}\}$ is bounded in ${Q_p}.$ Furthermore, $\{f_{w_n}\}$ converges to zero uniformly on every compact subset of ${\mathbb{D}}.$ Given a compact operator $K:{Q_p}\to{\mathcal{LF}(q, q-2, \frac{qp}{2})},$ by Lemma 3 we have $\lim_{n\to\infty}\|Kf_{w_n}\|_{{\mathcal{LF}(q, q-2, \frac{qp}{2})}} = 0.$ So

which implies that

It follows from Lemma 1 that

On the other hand, by Lemma 2, $T_{g_r}:{Q_p}\to {\mathcal{LF}(q, q-2, \frac{qp}{2})}$ is compact. Then

Using Lemma 1 again, we have

The proof is complete.

The following result can be deduced by Theorem 3 directly.

Corollary 2. Let $0 < p < 1$, $2 < q < \infty$ and $g\in H({\mathbb{D}})$. Then $T_g: \ {Q_p}\rightarrow{\mathcal{LF}(q, q-2, \frac{qp}{2})}$ is compact if and only if

4.   Conclusions

In this paper, we mainly prove that inclusion mapping $i: \ {Q_p}\rightarrow \mathcal{T}_{\frac{qp}{2}, s}^q(\mu)$ is bounded if and only if $\sup_{I\subseteq{\partial {\mathbb{D}}}}\frac{\left(\log\frac{2}{|I|}\right)^{q-s}\mu(S(I))}{|I|^{\frac{qp}{2}}} < \infty$, when $0 < p < 1$, $2 < q < \infty$ and $0 < s\leq q < \infty$. As an application, we prove that Volterra integral operator $T_g$ from $Q_p$ to the space $\mathcal{LF}(q, q-2, \frac{qp}{2})$ is bounded if and only if $g\in \mathcal{F}(q, q-2, \frac{qp}{2})$.

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, the Key Subject Program of Lingnan Normal University (No.1171518004) and (No.LZ1905), and Department of Education of Guangdong Province (No. 2018KTSCX133).

Conflict of interest

We declare that we have no conflict of interest.