Hankel Operators on the Weighted L P-Bergman Spaces with Exponential Type Weights

and Applied Analysis 3 for z ∈ D. Since h z is harmonic, there is an analytic function Φ z on D(z) such that Φ z (z) = 0 and ReΦ z = h z on D(z). Thus we have |ez | = ez . Hence by the submean value property together with Lemma 5, we get 󵄨󵄨󵄨󵄨f (z) 󵄨󵄨󵄨󵄨 p = 󵄨󵄨󵄨󵄨 f (z) e −(1/p)Φ


Introduction
Let D be the unit disc in the complex plane C and () the area measure on D, and denote by (D) the space of all analytic functions in D. Let  ∈  2 (D) with Δ > 0. For 0 <  ≤ ∞, the weighted Bergman space    is the space of functions  ∈  (D)  ( Note that    is the closed subspace of    :=   (D,  − ) consisting of analytic functions.Since the space  2   is a reproducing kernel Hilbert space, for each  ∈ D, there are functions   ∈  2   with () = ⟨,   ⟩, where ⟨⋅, ⋅⟩ is the usual inner product in  2   .The orthogonal projection from  2  to  2   is given by where (, ) =   ().
Given  ∈  1 (D) so that there exists a dense subset D of  2   with  ∈  2  for  ∈ D, the big Hankel operator   with symbol  is densely defined by where  is the orthogonal projection of  2  onto  2  .We write  = /.Then the -equation can be written by For  ∈  2  , we look for a solution V ∈  2  of minimal  2 norm.Notice that the solution of minimal norm is the one that is orthogonal to the kernel of  on  2  ; that is, V ⊥  2  .Then, if  ∈  2   solves (4), we get The linear operator  :  2  →  2  given by is called the canonical solution operator to  on  2  .For any  ∈  2   , obviously () =  and That is, the canonical solution operator coincides with the big Hankel operator acting on we consider the densely defined big Hankel operator on  2

𝜑
given by A positive function  on D is said to belong to the class E if it satisfies the following three properties.
(a) There exists a constant  1 > 0 such that (c) For each  ≥ 1, there are constants   > 0 and 0 <   < 1/ such that In this paper, we characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weighted   -Bergman spaces with exponential type weights as follows.
In [1], Luecking firstly proved the same results in the context of the ordinary  2 -Bergman spaces.For  2 -Bergman spaces with exponential type weights, the same results were proved in [2][3][4].Moreover, Schatten-class Hankel operators are also indicated in their papers.
The expression  ≲  means that there is a constant  independent of the relevant variables such that  ≤ , and  ≈  means that  ≲  and  ≲ .

Preliminaries
From now on we assume that  ∈  2 (D), Δ > 0, and the function () = (Δ()) −1/2 is in the class E. The following notations will be frequently used: where  1 and  2 are the constants in the conditions (a) and (b) in Section 1 and Lemma 3 (see [5]).For each  ≥ 1, there exists a constant  > 0 (depending on ) such that for ,  ∈ D, one has By using the upper estimates for (, ) in Lemma 3, Arroussi and Pau [5] proved that the orthogonal projection  projects    boundedly onto    for 1 ≤  ≤ ∞.
Lemma 4 (see [6]).Let 0 <  ≤   and  ∈ D.Then, By using the third Green formula, we get the following two approximation results.
Thus we have | Φ  | =  ℎ  .Hence by the submean value property together with Lemma 5, we get On the other hand, For | − | = (), we have Hence we have Thus Despite that the next result was proved in [4], we give the proof of different method by using Lemma 7.

Hankel Operators on 𝐴 𝑝 𝜑
For the proof of boundedness of Hankel operator on  2 -Bergman spaces with exponential type weights in [2][3][4], they used Hörmander's  2 -estimates for .However, for   -Bergman spaces, we need the following   -estimates for .