We generalize the notions of harmonic conjugate functions and Hilbert transforms to higher-dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These harmonic conjugates 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm193-2-4,
author = {Andreas Axelsson and Kit Ian Kou and Tao Qian},
title = {Hilbert transforms and the Cauchy integral in euclidean space},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {161-187},
zbl = {1168.45001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm193-2-4}
}
Andreas Axelsson; Kit Ian Kou; Tao Qian. Hilbert transforms and the Cauchy integral in euclidean space. Studia Mathematica, Tome 192 (2009) pp. 161-187. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm193-2-4/