Extended Cesaro operators between different Hardy spaces

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 Cesaro operators T(g)f(z) = integral(1)(0) f(tz)Rg(tz)dt/t and L(g)f(z) = integral(1)(0) Rf(tz)g(tz)dt/t, where z is an element of B and g is a fixed holomorphic map on B, acting from the space H-p into the space H-q, for the case p < q. Our results extend and simplify some one-dimensional results. (C) 2008 Elsevier Inc. All rights reserved.