Xiaofen Lv. Source: Annals of Functional Analysis, Volume 10, Number 1, 122--134.Abstract:
In this article, given some positive Borel measure $\mu$ , we define two integration operators to be
\[I_{\mu}(f)(z)=\int_{\mathbf{D}}f(w)K(z,w)e^{-2\varphi(w)}\,d\mu(w)\] and
\[J_{\mu}(f)(z)=\int_{\mathbf{D}}\vert f(w)K(z,w)\vert e^{-2\varphi(w)}\,d\mu(w).\] We characterize the boundedness and compactness of these operators from the Bergman space $A^{p}_{\varphi}$ to $L^{q}_{\varphi}$ for $1\lt p,q\lt \infty$ , where $\varphi$ 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 $A^{p}_{\varphi}$ to $L^{q}_{\varphi}(d\mu)$ , $0\lt p,q\lt \infty$ .