Measuring Triebel-Lizorkin fractional smoothness on domains in terms of first-order differences

TL;DR

This paper provides new equivalent characterizations of fractional Triebel-Lizorkin spaces on domains using first-order differences, applicable for non-integer smoothness levels and specific index conditions.

Contribution

It introduces a novel characterization of Triebel-Lizorkin spaces in terms of first-order differences for non-integer smoothness on uniform domains.

Findings

Characterizations valid for non-integer smoothness s>0.

Applicable for indices 1≤p<∞, 1≤q≤∞ with fractional part of s greater than d/p - d/q.

Extends understanding of fractional smoothness in domain settings.

Abstract

In this note we give equivalent characterizations for a fractional Triebel-Lizorkin space $F^s_{p,q}(\Omega)$ in terms of first-order differences in a uniform domain $\Omega$. The characterization is valid for any positive, non-integer real smoothness $s\in \mathbb{R}_+\setminus \mathbb{N}$ and {indices $1\leq p<\infty$, $1\leq q \leq \infty$} as long as the fractional part $\{s\}$ is greater than $d/p-d/q$.