General Toeplitz operators on weighted Bloch-type spaces in the unit ball of
© El-Sayed Ahmed; licensee Springer. 2013
Received: 17 October 2012
Accepted: 17 April 2013
Published: 9 May 2013
In this paper, we consider the weighted Bloch-type spaces with and in the unit ball of . We present some basic properties of the spaces , then we consider the Toeplitz operator acting between spaces, where μ is a positive Borel measure in the unit ball . Moreover, we characterize complex measures μ for which the Toeplitz operator is bounded or compact on .
MSC: 47B35, 32A18.
We start here with some terminology, notations and definitions of various classes of analytic functions defined on the unit ball of .
Thus , where 1 stands for a constant function.
Toeplitz operators have been studied extensively on the Bergman spaces by many authors. For references, see  and . Boundedness and compactness of the general Toeplitz operators on the α-Bloch spaces have been investigated in  on the unit disk for . Also, in , the authors extended the general Toeplitz operator to with . Recently, in , the general Toeplitz operators on the analytic Besov spaces with have been investigated. Under a prerequisite condition, the authors characterized a complex measure μ on the unit disk for which is bounded or compact on the Besov space . For more studies on the Toeplitz operator, we refer to [11–17].
In this paper, we consider the weighted Bloch-type spaces with and in the unit ball of . We prove a certain integral representation theorem that is used to determine the degree of growth of the functions in the space . It is also proved that the space is a Banach space for each weight , , and the Banach dual of the Bergman space is for each , . Further, we extend the Toeplitz operator to in the unit ball of and completely characterize the positive Borel measure μ such that is bounded or compact in spaces with .
Throughout the paper, we say that the expressions A and B are equivalent, and write whenever there exist positive constants and such that . As usual, the letter C denotes a positive constant, possibly different on each occurrence. Hereafter, ω stands for a right-continuous and nondecreasing function.
Theorem 1.1 (see [, Theorem 1.12])
have the following asymptotic properties.
(1) If , then and are both bounded in .
Lemma 1.1 (see [, Lemma 3.3])
Let be the Bergman metric on . Denote the Bergman metric ball at by , where and .
Lemma 1.2 (see [, Theorem 2.23])
For fixed , there is a sequence such that:
• there is a positive integer N such that each is contained in at most N of the sets .
then μ is called a vanishing Carleson measure.
If , , then we get the α-Bloch space and the little α-Bloch space . If , and , then we get the classical Bloch space and . These classes extend the weighted Bloch spaces defined in  to the setting of several complex variables.
If and , then we get the logarithmic α-Bloch space and the little logarithmic α-Bloch space . If , and , then we get the logarithmic Bloch space and (see ).
2 Holomorphic -Bloch space in the unit ball
In this section, we study the general -Bloch space in the unit ball of by giving some characterizations of -Bloch space, then we present several auxiliary results, which play important roles in the proofs of our main results.
from which the result follows. □
Theorem 2.1 For each , and . Then the following conditions are equivalent:
(ii) The function is bounded in ;
This proves that (i) ⇒ (ii).
As in the proof of Theorem 7.1 in , we have .
This shows that (ii) implies (iii). That (iii) implies (i) follows from differentiating under the integral sign and then applying Theorem 1.12 in . □
It remains for us to show that .
This shows that and completes the proof of the theorem. □
is bounded in .
It is easy to check that .
is bounded in . This completes the proof. □
Lemma 2.3 Let . Let λ be any real number satisfying the following properties:
• if ;
• if ;
• if .
This proves the necessity. The proof of the sufficiency condition is much akin to the corresponding one in , so the proof is omitted. □
Proposition 2.1 Suppose and . Let λ be any real number satisfying:
• if ;
• if .
3 Boundedness of general Toeplitz operators
In this section, we study the boundedness of general Toeplitz operators acting on the weighted Bloch-type spaces in the unit ball of .
Theorem 3.1 Let μ be a positive Borel measure on . Then we have
(1) if , then is bounded on if and only if and μ is a Carleson measure;
(2) if , then is bounded on if and only if and μ is a Carleson measure;
(3) if , , then is bounded on if and only if and μ is a Carleson measure.
for all and , where C is a positive constant that does not depend on f or g.
It is easy to see that
This implies that . Hence, is a bounded operator on with , .
Hence μ is a Carleson measure on .
By (8), then and .
When , taking , we have .
By (8) it is obvious that .
This completes the proof of Theorem 3.1. □
4 Compactness of general Toeplitz operators
In this section, we study the compactness of Toeplitz operators on the weighted Bloch-type spaces in the unit ball of . We need the following lemma.
whenever is a bounded sequence in that converges to 0 uniformly on .
Proof This lemma can be proved by Montel’s theorem and Lemma 2.1. □
Theorem 4.1 Let μ be a positive Borel measure on . We have the following:
(1) if , then is compact on if and only if and μ is a vanishing Carleson measure;
(2) if , then is compact on if and only if and μ is a vanishing Carleson measure;
(3) if , , then is compact on if and only if and μ is a vanishing Carleson measure.
From (8) it is easy to see that
which implies that is a compact operator.
This implies that μ is a vanishing Carleson measure on .
with , we have .
we have .
Then, it is obvious that .
This completes the proof of Theorem 4.1. □
Remark 4.1 It is still an open problem to study the properties of radial Toeplitz operators on the studied spaces of this paper. For more information on radial Toeplitz operators, we refer to [23, 24].
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