Abstract
In this paper, we consider Toeplitz operator acting on weighted Bloch spaces. Meanwhile, the inclusion map from weighted Bloch spaces into tent space is also investigated.
1. Introduction
Denote the open unit disk of the complex plane by and the boundary of by . Let denote the space of all functions analytic in . For any , is the automorphism of which exchanges for . Recall that is the Bergman metric. For any , is the Bergman disk. Let denote the normalized area of . From [1], we see that when is fixed.
For and , the weighted Bergman space is the space of all such that When , is the classical Bergman space. We refer the readers to [1, 2] for more results on weighted Bergman spaces.
Let . An is said to belong to the weighted Bloch space, denoted by , if The space has been studied extensively in [3]. See [1, 4ā8] for the study of some operators on weighted Bloch spaces.
Let . The Toeplitz operator with symbol is defined by where . There are many results related to , see [1] and the references therein. Especially, some characterizations for the operator on have been obtained by many authors. Since , it is nature to ask
The following theorem is the first main result in this paper.
Theorem 1. Let and be harmonic. Then the following statements hold.
(1) is bounded if and only if is bounded.
(2) is compact if and only if .
Given a positive Borel measure , the Toeplitz operator with the symbol is defined by
For the Toeplitz operator , we have the following result.
Theorem 2. Let and be a positive Borel measure. Then the following statements hold.
(1) is bounded if and only if (2) is compact if and only if
For , is the normalized length of the subarc and the corresponding Carleson square for is defined as follows (see [9]).
For , a positive Borel measure on is said to be a -Carleson measure ifIf , -Carleson measure is the classical Carleson measure. From Lemma 3.1.1 of [10], for we know that is a -Carleson measure if and only if Moreover, .
Let and be a positive Borel measure on . The tent space is the class of all which satisfy The tent space was introduced by J. Xiao [11] to studied Carleson measure for space. He proved that space is continuously contained in if and only if J. Pau and R. Zhao [12] generalized the main results in [11]. In [13], J. Liu and Z.Lou studied Morrey spaces. They proved that an equivalent condition for Morrey spaces continuously contained in is that is a Carleson measure. See [14, 15] for more information of the Morrey space.
We state the last main result in this paper as follows.
Theorem 3. Let and be a positive Borel measure. Then the following statements hold.
(1) The inclusion map is bounded if and only if is a -Carleson measure.
(2) The inclusion map is compact if and only if is a vanishing -Carleson measure.
Throughout this paper, the letter will denote constants and may differ from one occurrence to the other. The notation means that there is a positive constant C such that . The notation means and .
2. Proofs of Main Results
To prove our main results in this paper, we need some auxiliary results. The following result can be found in [16, Theorem 3.8].
Lemma 4. Let , , , and . Then if and only if
From Lemma 4, we can easily deduce the following result.
Lemma 5. Let , , , and . Then if and only if
Proof. First assume that . It is clear that Thus,The proof of the inverse direction is similar to the above statements we omit the details. The proof is complete.
Proof of Theorem 1. (1) First assume that . For , since we obtainHence is bounded.
Conversely, assume that is bounded. For , set It is easy to check that . Using Lemma 5 with , and , we getwhich implies that , as desired.
(2) Sufficiency. The result is obvious.
Necessity. For any , let . It is easy to check that uniformly on compact subsets on as From the fact that is compact on and the proof of (1), we haveBy the arbitrariness of and the Maximal Module Principle, we get The proof is complete.
Proof of Theorem 2. (1) First suppose that is bounded. For any , from the proof of Theorem 1, we obtain that and . Thus,as desired.
Conversely, suppose that Then we can get that is a Carleson measure for . If and , we can easily obtain that . Using Fubiniās Theorem, we obtainHence is bounded.
(2) Suppose that is compact. Let . Set Then and uniformly on compact subset on as Thus,which implies the desired result.
Conversely, assume that We know that is a vanishing Carleson measure for . We want to show that is compact. Using Fubiniās Theorem we have If and , we can easily obtain . Therefore,That is, If weakly in , it follows that . The proof of Theorem 2 is complete.
Proof of Theorem 3. (1) Suppose that is bounded. For , set Then . For any , we getas desired.
Conversely, assume that is a -Carleson measure. Let . Using the well-known fact we haveNote that Then is a Carleson measure for . Since , combined with Lemma 4, we obtainHence is bounded.
(2) Suppose that is compact. Let such that as . We know that and uniformly on compact subsets of as By Theorem 5.15 of [1] it follows that weakly as Hence for the compact operator , we have as Thus,as Hence is a vanishing -Carleson measure.
Conversely, assume that is a vanishing -Carleson measure. Let , , and āā uniformly on compact subsets of . Then it is easy to get thatLet and ; we get the desired result. The proof is complete.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
The author was partially supported by National Natural Science Foundation of China (no. 11801305).