The $L^p$-$L^q$ Boundedness and Compactness of Bergman Type Operators

Abstract

We investigate Bergman type operators on the complex unit ball, which are singular integral operators induced by the modified Bergman kernel. We consider the $L^p$-$L^q$ boundedness and compactness of Bergman type operators. The results of boundedness can be viewed as the Hardy–Littlewood–Sobolev (HLS) type theorem in the case unit ball. We also give some sharp norm estimates of Bergman type operators which in fact gives the upper bounds of the optimal constants in the HLS type inequality on the unit ball. Moreover, a trace formula is given.