A T(P) theorem for Zygmund spaces on domains

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 $D$.