Weighted composition operators on Bergman spaces Aωp

Let phi be an analytic self-map of the open unit disk D, and let u be an analytic function on D. The weighted composition operator induced by phi with weight u is given by (uC(phi)f)(z) = u(z)f(phi(z)) for z in D and f analytic on D. In this paper, we study weighted composition operators acting between two exponentially weighted Bergman spaces A(omega)(p) and A(omega)(q). We characterize the bounded, compact and Schatten class membership operators uC(phi) acting from A(omega)(p) to A(omega)(q) when 0 < p <= infinity and 0 < q < infinity. To obtain this, we first get an important estimate for the norm of the reproducing kernel in A(omega)(p) 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 and u = 1, we compare our criteria with those given by Kriete and MacCluer in [15].