Oscillations and differences in Triebel-Lizorkin-Morrey spaces

TL;DR

This paper introduces new characterizations of Triebel-Lizorkin-Morrey spaces on Lipschitz domains using local oscillations and higher order differences, extending classical function space theory.

Contribution

It provides novel equivalence characterizations of Triebel-Lizorkin-Morrey spaces via oscillations and differences, applicable on Lipschitz domains and general parameters.

Findings

New characterizations of Triebel-Lizorkin-Morrey spaces using local oscillations.

Characterizations involving higher order differences.

Applicable to classical Sobolev spaces as special cases.

Abstract

In this paper we are concerned with Triebel-Lizorkin-Morrey spaces $\mathcal{E}^{s}_{u,p,q}(\Omega)$ of positive smoothness $s$ defined on (special or bounded) Lipschitz domains $\Omega\subset\mathbb{R}^d$ as well as on $\mathbb{R}^d$. For those spaces we prove new equivalent characterizations in terms of local oscillations which hold as long as some standard conditions on the parameters are fulfilled. As a byproduct, we also obtain novel characterizations of $\mathcal{E}^{s}_{u,p,q}(\Omega)$ using differences of higher order. Special cases include standard Triebel-Lizorkin spaces $F^s_{p,q} (\Omega)$ and hence classical $L_p$-Sobolev spaces $H^s_p(\Omega)$. Key words: Triebel-Lizorkin-Morrey space, Morrey space, Lipschitz domain, oscillations, higher order differences