Weighted composition operators on Bergman spaces Aωp$A^p_\omega$.

Let ϕ be an analytic self‐map of the open unit disk D$\mathbb {D}$, and let u be an analytic function on D$\mathbb {D}$. The weighted composition operator induced by ϕ with weight u is given by (uCϕf)(z)=u(z)f(ϕ(z))$(uC_{\phi }f)(z)=u(z)f(\phi (z))$ for z in D$\mathbb {D}$ and f analytic on D$\mathbb {D}$. In this paper, we study weighted composition operators acting between two exponentially weighted Bergman spaces Aωp$A^p_{\omega }$ and Aωq$A^q_{\omega }$. We characterize the bounded, compact and Schatten class membership operators uCϕ$ uC_{\phi }$ acting from Aωp$A^p_{\omega }$ to Aωq$A^q_{\omega }$ when 0<p≤∞$ 0< p\le \infty$ and 0<q<∞$ 0< q<\infty$. To obtain this, we first get an important estimate for the norm of the reproducing kernel in Aωp$A^p_\omega$ and some new characterizations of Carleson measures. Our results use certain integral transforms that generalize the usual Berezin transform. In the case where p=q$p=q$ and u=1$u=1$, we compare our criteria with those given by Kriete and MacCluer in [15]. [ABSTRACT FROM AUTHOR]