Composition as an integral operator

Abstract

Let S be the unit sphere and B the unit ball in C-n, and denote by L-1 (S) the usual Lebesgue space of integrable functions on S. We define four "composition operators" acting on L-1 (S) and associated with a Borel function phi : S -> (B) over bar, by first taking one of four natural extensions of f is an element of L-1 (S) to a function on (B) over bar, then composing with phi and taking radial limits. Classical composition operators acting on Hardy spaces of holomorphic functions correspond to a special case. Our main results provide characterizations of when the operators we introduce are bounded or compact on L-t(S), 1 <= t < infinity. Dependence on t and relations between the characterizations for the different operators are also studied. (C) 2014 Elsevier Inc. All rights reserved.