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HILBERT TRANSFORMS AND THE CAUCHY INTEGRAL IN EUCLIDEAN SPACE

By Andreas Axelsson, Kit Ian Kou and Tao Qian

Abstract

Abstract. 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

Topics: Cauchy integral, Dirac equation, double layer potential, Hilbert transform, harmonic conjugates, Lipschitz domain MSC classes, 45E05, 31B10
OAI identifier: oai:CiteSeerX.psu:10.1.1.311.5834
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