Hilbertian Hardy-Sobolev spaces on a half-plane

Abstract

In this paper we deal with a scale of reproducing kernel Hilbert spaces $H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane $\mathbb{C}^+$. 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 $K_{z,n}$ of $H^{(n)}_2$, from which we can find the estimate $\Vert K_{z,n}\Vert\sim\vert z\vert^{-1/2}$ for $z\in\mathbb{C}^+$. Then composition operators $C_\varphi :H_2^{(n)} \to H_2^{(n)}$, $C_\varphi f=f\circ \varphi $, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps $\varphi: \mathbb{C}^+\to \mathbb{C}^+$ to induce bounded composition operators.