Smoothness of the Beurling transform in Lipschitz domains

Details Smoothness of the Beurling transform in Lipschitz domains Smoothness of the Beurling transform in Lipschitz domains

2012-01-25

arxiv.org/abs/1201.5385v1

Mathematics

Classical Analysis and ODEs

32 pages

Scientific paper

Let D be a planar Lipschitz domain and consider the Beurling transform of the characteristic function of D, B(1_D). Let 11. In this paper we show that if the outward unit normal N on bD, the boundary of D, belongs to the Besov space B_{p,p}^{a-1/p}(bD), then the Beurling transform of 1_D is in the Sobolev space W^{a,p}(D). This result is sharp. Further, together with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling transform is bounded in W^{a,p}(D) if N belongs to B_{p,p}^{a-1/p}(bD), assuming that ap>2.

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