Hilbertian Hardy-Sobolev spaces on a half-plane

Abstract

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

27 pp