Bergman spaces with exponential type weights

Abstract

Abstract For 1 ≤ p < ∞ $1\le p<\infty $ , let A ω p $A^{p}_{\omega }$ be the weighted Bergman space associated with an exponential type weight ω satisfying ∫ D | K z ( ξ ) | ω ( ξ ) 1 / 2 d A ( ξ ) ≤ C ω ( z ) − 1 / 2 , z ∈ D , $$ \int _{{\mathbb{D}}} \bigl\vert K_{z}(\xi ) \bigr\vert \omega (\xi )^{1/2} \,dA(\xi ) \le C \omega (z)^{-1/2}, \quad z\in {\mathbb{D}}, $$ where K z $K_{z}$ is the reproducing kernel of A ω 2 $A^{2}_{\omega }$ . This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂̅-equation (Theorem 2.5) for more general weight ω ∗ $\omega _{*}$ . As an application, we prove the boundedness of the Bergman projection on L ω p $L^{p}_{\omega }$ , identify the dual space of A ω p $A^{p}_{\omega }$ , and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from A ω p $A^{p}_{\omega }$ into A ω q $A^{q}_{\omega }$ , 1 ≤ p , q < ∞ $1\le p,q<\infty $ , such as Toeplitz and (big) Hankel operators.

Keywords

• Bergman space
• Bergman kernel
• Bergman projection
• Atomic decomposition