A T(P) theorem for Zygmund spaces on domains

TL;DR

This paper establishes a T(P) theorem for Zygmund spaces on Lipschitz domains, characterizing boundedness of Calderón-Zygmund operators via polynomial testing conditions, extending classical harmonic analysis results.

Contribution

It provides a novel T(P) theorem for Zygmund spaces on Lipschitz domains, linking operator boundedness to polynomial behavior, which was previously unexplored in this context.

Findings

Characterization of bounded operators on Zygmund spaces

Polynomial testing conditions for Calderón-Zygmund operators

Extension of T(P) theorems to Lipschitz domain settings

Abstract

Let $D\subset \mathbb{R}^d$ be a bounded Lipschitz domain, $\omega$ be a high order modulus of continuity and let $T$ be a convolution Calder\'{o}n-Zygmund operator. We characterize the bounded restricted operators $T_D$ on the Zygmund space $\mathcal{C}_{\omega}(D)$. The characterization is based on properties of $T_D P$ for appropriate polynomials $P$ restricted to $D$.