General Toeplitz operators on weighted Bloch-type spaces in the unit ball of
Journal of Inequalities and Applications volume 2013, Article number: 237 (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 .
Let be the unit ball of the n-dimensional complex Euclidean space . The boundary of is denoted by and is called the unit sphere in . Occasionally, we will also need the closed unit ball . We denote the class of all holomorphic functions on the unit ball by . The ball centered at with radius r is denoted by . For , let , where dν is the normalized Lebesgue volume measure on and (where Γ denotes the gamma function) so that . The surface measure on is denoted by dσ. Once again, we normalize σ so that . For any , the inner product is defined by
For every point , the Möbius transformation is defined by
where , , and . The map has the following properties that , , and
where z and w are arbitrary points in . In particular,
For , the holomorphic gradient of f at z is defined by
and the radial derivative of f at z is defined by
Similarly, the Möbius invariant complex gradient of f at z is defined by
For , a function is said to belong to the α-Bloch spaces if (see )
The little Bloch space consists of all such that
With the norm , we know that becomes a Banach space and is its closed subspace (see ). For , the spaces and become the Bloch and the little Bloch space, respectively (see, for example, [2–5]). Zhu in  says that the norm is equivalent to
For and , the weighted Bergman space consists of holomorphic functions such that
that is, . When the weight , we simply write for . In the special case when , is a Hilbert space. It is well known that for , the Bergman kernel of is given by
For , a complex measure μ is such that
The general Bergman projection is the orthogonal projection of the measure μ from into defined by
The general Bergman projection of the function f is
Let be a right-continuous and nondecreasing function. For a complex measure , , and , define weighted general Toeplitz operator as follows:
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])
Suppose b is real and . Then the integrals
have the following asymptotic properties.
(1) If , then and are both bounded in .
(2) If , then
(3) If , then
Lemma 1.1 (see [, Lemma 3.3])
Suppose γ is a real constant and . If
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 .
A positive Borel measure μ on the unit ball is said to be a Carleson measure for the Bergman space if
It is well known that a positive Borel measure μ is a Carleson measure if and only if there is a positive constant C such that
where is the sequence in Lemma 1.2. If μ satisfies that
then μ is called a vanishing Carleson measure.
For a given reasonable function , the weighted Bloch space of several complex variables is defined as the set of all analytic functions f on satisfying
for some fixed . In the special case where , reduces to the classical Bloch space ℬ in . This class of functions extends and generalizes the well known Bloch space. Now, we define the space in the unit ball . For and , a function is said to belong to the -Bloch space if
The little -Bloch space is a subspace of consisting of all such that
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.
The logarithmic -Bloch space is the space of holomorphic functions f such that
Correspondingly, the little logarithmic -Bloch space is a subspace of consisting of all functions f such that
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.
Lemma 2.1 Let and . Suppose that
Proof Let , and . By the definition of and , we have that
Since and , , we get
from which the result follows. □
Theorem 2.1 For each , and . Then the following conditions are equivalent:
(ii) The function is bounded in ;
(iii) There exists a function such that
Proof By the Cauchy-Schwarz inequality in , we have
This proves that (i) ⇒ (ii).
If (ii) holds, then the function
is bounded in . For consider the holomorphic function
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 . □
Theorem 2.2 For each , , and . If , then the Banach dual of can be identified with (with equivalent norms) under the following integral pairing:
Proof It is easy to see that . If , then by Theorem 2.1, there exists a function such that
and , where C is a positive constant independent of g. By Fubini’s theorem,
Applying Lemma 2.15 in  for all , we have
Combining this, we see that
Conversely, if F is a bounded linear functional on and , then
It is easy to verify (using the homogeneous expansion of the kernel function) that
Define a function g on by
It remains for us to show that .
We interchange differentiation and the application of F, which can be justified by using the homogeneous expansion of the kernel. The result is
Since F is bounded on , we have
An application of Theorem 1.1 for then shows that
This shows that and completes the proof of the theorem. □
Lemma 2.2 If , , then if and only if the function
is bounded in .
Proof Recall from Lemma 2.14 in  that
So, the boundedness of
implies that of
On the other hand, if , then by Theorem 2.1,
where g is a function in . Now we let , where and
An application of Lemma 1.1 gives
Since is bounded, by Theorem 1.1 we have
It is easy to check that .
Using the product rule, we have
and we have
is bounded in . This completes the proof. □
Lemma 2.3 Let . Let λ be any real number satisfying the following properties:
• if ;
• if ;
• if .
Then, for all a holomorphic function if and only if
Proof Let . By a similar proof to the one for Theorem 3.1 in , we have
for any with . We know that
Thus, there is a constant such that
If , then
Now suppose as in , there is a constant such that this integral in (4) is dominated by
Combining with (5), we get that whenever ,
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 .
Proof Let in (3), then we have
Now, replacing f by , we get
by Lemma 2.2, we obtain that
Letting and in (7), we obtain
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.
Proof Since the Banach dual of is under the pairing (2), to prove the boundedness of , it suffices to show that
for all and , where C is a positive constant that does not depend on f or g.
For , by Fubini’s theorem, we get
Using the operator , we have
where , and I is the identity operator. Also,
By Proposition 2.1, we have
Next considering , we have
As in , by simple calculation, we have
It is easy to see that
(1) if and , then
(2) if , , then
(3) if , , and , then
This implies that . Hence, is a bounded operator on with , .
Conversely, suppose that is a bounded operator on . Take
It is clear that . On the other hand, take
Then, we have . Therefore
for every . This implies that
Hence μ is a Carleson measure on .
From the proof of the sufficient condition, we find that there exists a constant C such that
This implies that
If , we have
Take ; and . It is clear that . Taking , then
From (8) we have . Let , we have
Take ; and . It is clear that
Taking , then
By (8), then and .
When , taking , we have .
From Lemma 2.1, we get
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.
Lemma 4.1 Let , and be a bounded linear operator from into . When , and , then is compact if and only if
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.
Proof For , let be a sequence in satisfying and converges to 0 uniformly as on . Suppose . By duality, we have that is compact on if and only if
Similarly, as in the proof of Theorem 3.1 for , we have
For fixed , since μ is a vanishing Carleson measure, there exists such that
where and . For a positive constant , as in the proof of Theorem 3.1, by Proposition 2.1, we obtain
Since as on compact subsets of , we can choose j big enough so that
Now, taking δ such that , then
Hence , where C does not depend on , and so
Thus, is compact on if and only if
Again, as in the proof of Theorem 3.1, we have
From (8) it is easy to see that
(1) if and , then
(2) if , , then
(3) if , , and , then
Combined with as on compact subsets of , we have
which implies that is a compact operator.
Next assume that is a compact operator on . Again, as in the proof of Theorem 3.1, we take
We know that . On the other hand, take
Then and uniformly on compact subsets of , as ,