Extended Cesàro operators between different Hardy spaces

Abstract

Let Hp denote the Hardy space of holomorphic functions on the unit ball B. This note gives some sufficient and necessary conditions for the boundedness and compactness of the following extended Cesàro operatorsTg f (z) = ∫01 f (tz) R g (tz) frac(dt, t) and Lg f (z) = ∫01 R f (tz) g (tz) frac(dt, t),where z ∈ B and g is a fixed holomorphic map on B, acting from the space Hp into the space Hq, for the case p < q. Our results extend and simplify some one-dimensional results.