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We generalize the notion of harmonic conjugate functions and Hilberttransforms to higher dimensional euclidean spaces, in the setting ofdifferential forms and the Hodge-Dirac system. These conjugate functions are ingeneral far from being unique, but under suitable boundary conditions we proveexistence and uniqueness of conjugates. The proof also yields invertibilityresults for a new class of generalized double layer potential operators onLipschitz surfaces and boundedness of related Hilbert transforms.

Hilbert transforms and the Cauchy integral in euclidean space