Restricted Beurling transforms on Campanato spaces

Abstract

Let $\Omega \subset \mathbb{C}$ be a bounded domain with ${\mathcal{C}}^{1,\omega }$-smooth boundary, where $\omega$ is a Dini-smooth modulus of continuity. We prove that the restricted Beurling transform is bounded on the Campanato space ${\mathrm{BMO}}_{\omega }(\Omega )$.

12 pages, 1 figure