Toeplitz Operator and Carleson Measure on Weighted Bloch Spaces

Denote the open unit disk of the complex plane C by D and the boundary of D by ∂D. Let H(D) denote the space of all functions analytic in D. For any a ∈ D, φa (z) = a − z 1 − az , z ∈ D (1) is the automorphism ofDwhich exchanges 0 for a. Recall that β (z, a) = 12 log1 + 󵄨󵄨󵄨󵄨φa (z)󵄨󵄨󵄨󵄨 1 − 󵄨󵄨󵄨󵄨φa (z)󵄨󵄨󵄨󵄨 (2) is the Bergman metric. For any 0 < r < ∞, a ∈ D, D (a, r) = {z ∈ D : β (z, a) < r} (3) is the Bergman disk. Let |D(a, r)| denote the normalized area ofD(a, r). From [1], we see that |D(a, r)| ≈ (1 − |a|2)2 when r is fixed. For 0 < p < ∞ and α > −1, the weighted Bergman space Apα is the space of all f ∈ H(D) such that 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩pApα fl ∫D 󵄨󵄨󵄨󵄨f (z)󵄨󵄨󵄨󵄨p (1 − |z|2)α dA (z) < ∞. (4) When α = 0, Apα is the classical Bergman space. We refer the readers to [1, 2] for more results onweighted Bergman spaces. Let 0 < α < ∞. An f ∈ H(D) is said to belong to the weighted Bloch space, denoted byB, if 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Bα = sup z∈D (1 − |z|2)α 󵄨󵄨󵄨󵄨󵄨f󸀠 (z)󵄨󵄨󵄨󵄨󵄨 < ∞. (5) The spaceB has been studied extensively in [3]. See [1, 4–8] for the study of some operators on weighted Bloch spaces. Let φ ∈ L∞(D). The Toeplitz operator Tφ with symbol φ is defined by Tφf (z) = ∫ D φ (w) f (w) (1 − wz)2+α dAα (w) , (6)


Introduction
When  = 0,    is the classical Bergman space.We refer the readers to [1,2] for more results on weighted Bergman spaces.
Let  ∈  ∞ (D).The Toeplitz operator   with symbol  is defined by where   () = (1 − || 2 )  ().There are many results related to   , see [1] and the references therein.Especially, some characterizations for the operator   on  2  have been obtained by many authors.Since B  ⊆  1   , it is nature to ask The following theorem is the first main result in this paper.
Given a positive Borel measure , the Toeplitz operator with the symbol  is defined by For the Toeplitz operator   , we have the following result.
For 0 <  < ∞, a positive Borel measure  on D is said to be a -Carleson measure if If  = The tent space  ,2  was introduced by J. Xiao [11] to studied Carleson measure for   space.He proved that   space is continuously contained in  ,2   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  2, continuously contained in  ,2  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 0 <  < ∞ and  be a positive Borel measure.Then the following statements hold.
(1) The inclusion map   : (2) The inclusion map   : 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  ≲ .

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].
From Lemma 4, we can easily deduce the following result.
Then  ∈ B +1 if and only if Proof.First assume that  ∈ B +1 .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 as desired.
Conversely, suppose that sup ∈D  ( (, )) Then we can get that  is a Carleson measure for Hence Then    ∈ B +1 and    → 0 uniformly on compact subset on D as |  | → 1.Thus, which implies the desired result.Conversely, assume that lim We know that  is a vanishing Carleson measure for  1 (  ).