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

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 $sp>d$.