A T(P) theorem for Zygmund spaces on domains

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 Calderón-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 $D$.

19 pages; minor changes and corrections