Hilbertian Hardy-Sobolev spaces on a half-plane

Abstract

In this paper we deal with a scale of reproducing kernel Hilbert spaces H2(n), n≥0, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane C+. They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel Kz,n of H2(n), from which we can find the estimate ‖Kz,n‖∼|z|−1/2 for z∈C+. Then composition operators Cφ:H2(n)→H2(n), Cφf=f∘φ, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps φ:C+→C+ to induce bounded composition operators.