On a class of generalized Berezin type operators on the unit ball of Cn

Abstract

We characterize boundedness of a class of generalized Berezin type operators $S_{\mu }^{s,\beta ,r}$ from a weighted Bergman space to a weighted Lebesgue space on the unit ball of $C^n$. These types of operators are crucial in characterizing Carleson measures through products of functions. By an improved method using Khinchine's inequality and Khinchine-Kahane-Kalton inequality, our result not only extends a previous result on these operators to a large scale, but also removes an extra restrictive condition to that result. When $(s,\beta ,r)=(0,\beta ,1)$, our result also recovers a recent result on boundedness of Berezin type operators. As an application, we provide an alternate proof of a necessary condition for the parameter c in the $L_{\alpha }^p-L_{\beta }^q$ boundedness of the Forelli-Rudin type operator $S_{a,b,c}$.