Hilbert transforms and the Cauchy integral in euclidean space

Details Hilbert transforms and the Cauchy integral in euclidean space Hilbert transforms and the Cauchy integral in euclidean space

2008-09-24

arxiv.org/abs/0809.4128v3

Mathematics

Analysis of PDEs

Some minor corrections made

Scientific paper

We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert transforms.

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