Sarason's composition operator over the half-plane

Abstract

Let H={z∈C:Imz>0} be the upper half plane, and denote by Lp(R), 1≤p<∞, the usual Lebesgue space of functions on the real line R. We define two “composition operators” acting on Lp(R) induced by a Borel function φ:R→H‾, by first taking either the Poisson or Borel extension of f∈Lp(R) to a function on H‾, then composing with φ and taking vertical limits. Classical composition operators, induced by holomorphic functions and acting on the Hardy spaces Hp(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 Lp(R), 1≤p<∞. The characterization for the case 1<p<∞ is independent of p and the same for the Poisson and the Borel extensions. The case p=1 is quite different.