Restricted Beurling transforms on Campanato spaces

TL;DR

This paper proves that the restricted Beurling transform is bounded on Campanato spaces with Dini-smooth boundary conditions, extending understanding of singular integral operators in complex analysis.

Contribution

It establishes the boundedness of the restricted Beurling transform on Campanato spaces for domains with Dini-smooth boundaries, a novel extension in the theory of singular integrals.

Findings

Boundedness of the restricted Beurling transform on Campanato spaces.

Extension of singular integral operator theory to Dini-smooth boundary domains.

Advancement in understanding complex analysis operators on smooth boundary domains.

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 $\mathrm{BMO}_\omega(\Omega)$.