Composition as an integral operator

Let S be the unit sphere and B the unit ball in Cⁿ, and denote by L¹(S) the usual Lebesgue space of integrable functions on S. We define four "composition operators" acting on L¹(S) and associated with a Borel function ϕ:S→B-, by first taking one of four natural extensions of 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 L^t(S), 1≤t<∞. Dependence on t and relations between the characterizations for the different operators are also studied.