LAUSR

Abstract Let H = { z ∈ C : Im z > 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.