T1 theorem for Campanato spaces on domains

TL;DR

This paper establishes a T1 theorem characterizing when Calderón-Zygmund operators map Campanato spaces on Lipschitz domains into themselves, with conditions depending on the boundary smoothness and the operator's kernel.

Contribution

It provides a necessary and sufficient T1 condition for the boundedness of restricted Calderón-Zygmund operators on Campanato spaces on Lipschitz domains.

Findings

The T1 condition involves the characteristic function of the domain.

Boundedness holds for even kernel Calderón-Zygmund operators on $C^{1,\tilde{\omega}}$-smooth domains.

The results are sharp and depend on boundary regularity and kernel symmetry.

Abstract

Given a Lipschitz domain $D\subset \mathbb{R}^d,$ a Calder\'on-Zygmund operator $T$ and a modulus of continuity $\omega(x),$ we solve a problem when the restricted operator $T_Df=T(f\chi_D)\chi_D$ sends the Campanato space $\mathcal{C}_\omega(D)$ into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function $\chi_D$ of $D$: $$(T\chi_D)\chi_D \in \mathcal{C}_{\tilde{\omega}}(D),$$ assumed $\tilde{\omega}(x)= \omega(x)/\int_x^1 \omega(t)dt/t.$ To check the hypotheses of T1 theorem we need extra restrictions on both the boundary of $D$ and the operator $T.$ It is proved that the restricted Calder\'on-Zygmund operator $T_D$ with the even kernel is bounded on $\mathcal{C}_\omega(D),$ provided $D$ be $C^{1,\tilde{\omega}}-$smooth domain. This result is sharp.