Bergman spaces with exponential type weights

Abstract

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