Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions

Abstract

We consider Schatten class membership of Hankel operators on Paley--Wiener spaces of convex $\Omega \subset {\mathbb{R}}^n$, both for bounded and unbounded domains. In particular, the classical product Hardy spaces fit within our theory. For admissible domains, we develop a framework and theory of Besov spaces of Paley--Wiener type, and prove that a Hankel operator belongs to the Schatten class $S^p$ if and only if its symbol belongs to a corresponding Besov space, for $1\le p\le 2$. We extend this result to all $1\le p<\infty$ for the classical product Hardy spaces and to $1\le p<2(n+1)/(n-1)$ for the Paley--Wiener space of a bounded smooth domain $\Omega \subset {\mathbb{R}}^n$ of strictly positive curvature.

This article supersedes a previous preprint of the authors. The main results now characterize the Schatten class Hankel operators on the product Hardy space as well as for the Paley Wiener space associated with any bounded strongly convex set in Rn