Oscillations and differences in Besov-Morrey and Besov-type spaces

TL;DR

This paper develops new characterizations of Besov-Morrey and Besov-type spaces on Lipschitz domains using local oscillations and higher order differences, extending classical Besov space results.

Contribution

It introduces novel characterizations of Besov-Morrey and Besov-type spaces via oscillations and differences, combining Hedberg-Netrusov and Triebel's approaches.

Findings

New oscillation-based characterizations of function spaces.

Extension of classical Besov space results to Besov-Morrey and Besov-type spaces.

Applicable on Lipschitz domains with standard parameter conditions.

Abstract

In this paper we investigate Besov-Morrey spaces $\mathcal{N}^{s}_{u,p,q}(\Omega)$ and Besov-type spaces $B^{s,\tau}_{p,q}(\Omega)$ of positive smoothness defined on Lipschitz domains $\Omega \subset \mathbb{R}^d$ as well as on $\mathbb{R}^d$. We combine the Hedberg-Netrusov approach to function spaces with distinguished kernel representations due to Triebel, in order to derive novel characterizations of these scales in terms of local oscillations provided that some standard conditions concerning the parameters are fulfilled. In connection with that we also obtain new characterizations of $\mathcal{N}^{s}_{u,p,q}(\Omega)$ and $B^{s,\tau}_{p,q}(\Omega)$ via differences of higher order. By the way we recover and extend corresponding results for the scale of classical Besov spaces $B^{s}_{p,q}(\Omega)$. Key words: Besov-Morrey space, Besov-type space, Morrey space, Lipschitz domain, oscillations, higher order differences