Besov spaces on open sets

TL;DR

This paper defines and explores properties of Besov spaces on arbitrary open sets in , using spectral theory of SchrF6dinger operators with Dirichlet boundary conditions, establishing fundamental relations and isomorphisms.

Contribution

It introduces a new framework for Besov spaces on open sets via spectral theory, including test function spaces and isomorphism relations with SchrF6dinger operators.

Findings

Defined Besov spaces on open sets using spectral theorem

Established embedding relations and duality properties

Proved isomorphism relations among Besov spaces

Abstract

This paper is devoted to giving definitions of Besov spaces on an arbitrary open set of $\mathbb R^n$ via the spectral theorem for the Schr\"odinger operator with the Dirichlet boundary condition. The crucial point is to introduce some test function spaces on $\Omega$. The fundamental properties of Besov spaces are also shown, such as embedding relations and duality, etc. Furthermore, the isomorphism relations are established among the Besov spaces in which regularity of functions is measured by the Dirichlet Laplacian and the Schr\"odinger operators.