Zygmund regularity of even singular integral operators on domains

Given a bounded Lipschitz domain $D\subset {\mathbb{R}}^d,$ a convolution Calderón-Zygmund operator $T$ and a growth function $\omega (x)$ of type $n$, we study what conditions on the boundary of the domain are sufficient for boundedness of the restricted even operator $T_D$ on the generalized Zygmund space $C_{\ast }^{\omega }(D)$. Based on a recent T(P) theorem, we prove that this holds if the smoothness of the boundary of a domain $D$ is by one point, in a sense, greater than the smoothness of the corresponding Zygmund space $C_{\ast }^{\omega }(D)$. The main argument of the proof are the higher order gradient estimates of the transform $T_D{\chi }_D$ of the characteristic function of a domain with the polynomial boundary.