On the Π-operator in Clifford analysis

In this paper we prove that a generalization of complex Π-operator in Clifford analysis, obtained by the use of two orthogonal bases of a Euclidean space, possesses several mapping and invertibility properties, as studied before for quaternion-valued functions as well as in the standard Clifford ana...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2016-02, Vol.434 (2), p.1138-1159
Main Authors: Abreu Blaya, Ricardo, Bory Reyes, Juan, Guzmán Adán, Alí, Kähler, Uwe
Format: Article
Language:English
Subjects:
Beltrami equation
Clifford analysis
Pi-operator
Teodorescu transform
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Summary:In this paper we prove that a generalization of complex Π-operator in Clifford analysis, obtained by the use of two orthogonal bases of a Euclidean space, possesses several mapping and invertibility properties, as studied before for quaternion-valued functions as well as in the standard Clifford analysis setting. We improve and generalize most of those previous results in this direction and additionally other consequent results are presented. In particular, the expression of the jump of the generalized Π-operator across the boundary of the domain is obtained as well as an estimate for the norm of the Π-operator is given. At the end an application of the generalized Π-operator to the solution of Beltrami equations is studied where we give conditions for a solution to realize a local and global homeomorphism.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2015.09.038