Composition as an integral operator

Let S be the unit sphere and B the unit ball in CnCn, and denote by L1(S)L1(S) the usual Lebesgue space of integrable functions on S. We define four “composition operators” acting on L1(S)L1(S) and associated with a Borel function φ:S→B¯, by first taking one of four natural extensions of f∈L1(S)f∈L1(S) to a function on B¯, then composing with φ   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 Lt(S)Lt(S), 1≤t<∞1≤t<∞. Dependence on t and relations between the characterizations for the different operators are also studied.