Interpolation spaces of generalized smoothness and their applications to elliptic equations

TL;DR

This paper introduces new classes of generalized smoothness spaces via interpolation methods, broadening classical function spaces and applying them to elliptic equations on manifolds.

Contribution

It develops a framework of generalized smoothness spaces using interpolation, extending classical spaces to manifolds and applying them to elliptic problems.

Findings

New interpolation-based smoothness spaces introduced

Spaces applicable to elliptic equations on manifolds

Generalizations of Sobolev, Nikolskii-Besov, Triebel-Lizorkin spaces

Abstract

We introduce and investigate classes of normed or quasinormed distribution spaces of generalized smoothness that can be obtained by various interpolation methods applied to classical Sobolev, Nikolskii-Besov, and Triebel-Lizorkin spaces. An arbitrary positive function O-regularly varying at infinity serves as the order of regularity for the spaces introduced. They are broad generalizations of the above classical spaces and allow being well defined on smooth manifolds. We give applications of the spaces under investigation to elliptic equations and elliptic problems on smooth manifolds.