Geodesic
Hilbert transforms and the Cauchy integral in euclidean space
Andreas Axelsson 1   ; Kit Ian Kou 2   ; Tao Qian 2
1 Matematiska Institutionen Stockholms Universitet 106 91 Stockholm, Sweden
2 Department of Mathematics University of Macau Taipa, Macau, China
Studia Mathematica, Tome 193 (2009) no. 2, pp. 161-187 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/sm193-2-4
Keywords: generalize notions harmonic conjugate functions hilbert transforms higher dimensional euclidean spaces setting differential forms hodge dirac system these harmonic conjugates general far being unique under suitable boundary conditions prove existence uniqueness conjugates proof yields invertibility results class generalized double layer potential operators lipschitz surfaces boundedness related hilbert transforms

Andreas Axelsson  1   ; Kit Ian Kou  2   ; Tao Qian  2

1 Matematiska Institutionen Stockholms Universitet 106 91 Stockholm, Sweden
2 Department of Mathematics University of Macau Taipa, Macau, China 
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Andreas Axelsson; Kit Ian Kou; Tao Qian. Hilbert transforms and the Cauchy integral
 in euclidean space. Studia Mathematica, Tome 193 (2009) no. 2, pp. 161-187. doi: 10.4064/sm193-2-4

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