  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 $\partial D$ 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.
