A T(1) theorem for fractional Sobolev spaces on domains

TL;DR

This paper establishes a T(1)-theorem for fractional Sobolev spaces on uniform domains, providing a new characterization of these spaces and extending boundedness results for Calderón-Zygmund operators.

Contribution

It introduces a novel norm characterization for fractional Sobolev spaces on uniform domains and proves a T(1)-theorem for a broad class of operators in this setting.

Findings

Characterization of fractional Sobolev spaces via first order differences.

Proof of a T(1)-theorem for these spaces on uniform domains.

Boundedness of Calderón-Zygmund operators under new conditions.

Abstract

Given any uniform domain $\Omega$, the Triebel-Lizorkin space $F^s_{p,q}(\Omega)$ with $0<s<1$ and $1<p,q<\infty$ can be equipped with a norm in terms of first order differences restricted to pairs of points whose distance is comparable to their distance to the boundary. Using this characterization, originally due to Seeger and reproven here, we prove a T(1)-theorem for fractional Sobolev spaces with $0<s<1$ for any uniform domain and for a large family of Calder\'on-Zygmund operators in any ambient space $\mathbb{R}^d$ as long as $sp>d$.