Embedding theorems and integration operators on Bergman spaces with exponential weights

Abstract

In this article, given some positive Borel measure $\mu$, we define two integration operators to be

$$I_{\mu }(f)(z)={\int }_{\mathbf{D}}f(w)K(z,w)e^{-2\phi (w)}d\mu (w)$$ and

$$J_{\mu }(f)(z)={\int }_{\mathbf{D}}|f(w)K(z,w)|e^{-2\phi (w)}d\mu (w).$$ We characterize the boundedness and compactness of these operators from the Bergman space $A_{\phi }^p$ to $L_{\phi }^q$ for $1<p,q<\infty$, where $\phi$ belongs to a large class ${\mathcal{W}}_0$, which covers those defined by Borichev, Dhuez, and Kellay in 2007. We also completely describe those $\mu$’s such that the embedding operator is bounded or compact from $A_{\phi }^p$ to $L_{\phi }^q(d\mu )$, $0<p,q<\infty$.