Sarason's composition operator over the half-plane

Let H={z∈C:Imz>0} be the upper half plane, and denote by L^p(R), 1≤p<∞ 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 φ:R→H‾ by first taking either the Poisson or Borel extension of f∈L^p(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 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<∞. 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.