Geodesic
Regularity of the Beurling transform in smooth domains
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 57-67

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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. 
@article{ZNSL_2015_434_a4,
     author = {A. V. Vasin},
     title = {Regularity of the {Beurling} transform in smooth domains},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {57--67},
     publisher = {mathdoc},
     volume = {434},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_434_a4/}
}
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A. V. Vasin. Regularity of the Beurling transform in smooth domains. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 57-67. http://geodesic.mathdoc.fr/item/ZNSL_2015_434_a4/