Singular integral operators on Zygmund spaces on domains

Given a bounded Lipschitz domain $D\subset \mathbb{R}^d$ and a Calderón–Zygmund operator $T$, we study the relationship between smoothness properties of $\partial D$ and the boundedness of $T$ on the Zydmund space $\mathcal{C}_{\omega}(D)$ defined for a general growth function $\omega$. We prove a T(P)-theorem for the Zygmund spaces, checking the boundedness of $T$ on a finite collection of polynomials restricted to the domain. Also, we obtain a new form of an extra cancellation property for the even Calderón–Zygmund operators in polynomial domains.