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

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 $\mathrm{\partial }D$ 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.