A $T(P)$-Theorem for Zygmund Spaces on Domains

Suppose given a bounded Lipschitz domain $D\subset \mathbb{R}^d$, a higher-order modulus of continuity $\omega$, and a convolution Calderón–Zygmund operator $T$. The restricted operators $T_D$ that are bounded on the Zygmund space $\mathcal{C}_{\omega}(D)$ are described. The description is based on properties of the functions $T_D P$ for appropriate polynomials $P$ restricted to $D$.