44 research outputs found
Hilbert transforms and the Cauchy integral in euclidean space
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.Comment: Some minor corrections mad