Sarason's composition operator over the half-plane

Abstract

Let H = {z is an element of C : Im z > 0} be the upper half plane, and denote by L-p(R), 1 <= p < infinity, the usual Lebesgue space of functions on the real line R. We define two "composition operators" acting on L-p(R) induced by a Borel function phi : R -> <(H)over bar> by first taking either the Poisson or Borel extension of f is an element of L-p(R) to a function on (H) over bar, then composing with phi and taking vertical limits. Classical composition operators, induced by holomorphic functions and acting on the Hardy spaces H-p(H) 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-p(R), 1 <= p < infinity The characterization for the case 1 < p < infinity is independent of p and the same for the Poisson and the Borel extensions. The case p = 1 is quite different. (C) 2018 Elsevier Inc. All rights reserved.