Restricted Beurling transforms on Campanato spaces

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

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