Calder\'on-Zygmund operators on Zygmund spaces on domains

TL;DR

This paper investigates how the smoothness of domain boundaries affects the boundedness of Calderón-Zygmund operators on Zygmund spaces, introducing a T(P) theorem and new cancellation properties.

Contribution

It establishes a T(P) theorem for Zygmund spaces on Lipschitz domains and introduces a novel cancellation property for even Calderón-Zygmund operators in polynomial domains.

Findings

Boundedness of Calderón-Zygmund operators linked to boundary smoothness.

Development of a T(P) theorem involving polynomials on domains.

Identification of a new cancellation property for even operators.

Abstract

Given a bounded Lipschitz domain $D\subset \mathbb{R}^d$ and a Calder\'on-Zygmund operator $T$, we study the relations between smoothness properties of $\partial D$ and the boundedness of $T$ on the Zydmund space $\mathcal{C}_{\omega}(D)$ defined for a general growth function $\omega$. In the proof we obtain a T(P) theorem for the Zygmund spaces, when one checks boundedness not only of the characteristic function, but a finite collection of polynomials restricted to the domain. Also, a new form of extra cancellation property of the even Calder\'on-Zygmund operators in polynomial domains is stated.