Abstract:
Let $D\subset \mathbb{R}^d$ be a bounded Lipschitz domain, let $\omega$ be a modulus of continuity of a high order of smoothness, and let $T$ be a Calderón–Zygmund convolution operator with even kernel. Using the recent $T(P)$ boundedness criterion found by the author with E. Doubtsov, we prove that the operator $T$ is bounded in the Zygmund space $\mathcal{C}_{\omega}(D)$ if the smoothness of the boundary of $D$ is by one higher than that of $\mathcal{C}_{\omega}(D)$. The proof is based on estimates for the potentials with Calderón–Zygmund kernels of the characteristic function of a domain with a polynomial boundary.
Keywords:Zygmund spaces on smooth domains, Calderón–Zygmund operators with even
kernel.
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