Regularity of the Beurling transform in smooth domains

The relationship between smoothness properties of the boundary of a domain $\Omega$ and the boundedness of the Beurling transform in the corresponding Lipschitz classes $\mathrm{Lip}(\omega)$ for the case of a Dini-regular modulus of continuity $\omega$ is studied. The result is sharp. Our motivation arises from the work of Mateu, Orobitg and Verdera.