The Higher Spin Laplace Operator

This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a generalisation of the Laplace operator to higher spin as we...

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Bibliographic Details
Published in:Potential analysis 2017-08, Vol.47 (2), p.123-149
Main Authors: De Bie, Hendrik, Eelbode, David, Roels, Matthias
Format: Article
Language:English
Subjects:
Differential equations
Ellipticity
Functional Analysis
Functions (mathematics)
Geometry
Harmonics
Mathematical analysis
Mathematics
Mathematics and Statistics
Nonlinear programming
Operators (mathematics)
Polynomials
Potential Theory
Probability Theory and Stochastic Processes
Representations
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Summary:This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a generalisation of the Laplace operator to higher spin as well as a second-order analogue of the Rarita-Schwinger operator. To construct these operators, we will use the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the orthogonal group can be realised in terms of polynomials satisfying a system of differential equations. As a consequence, the functions on which this particular class of operators act are functions taking values in the space of harmonics homogeneous of degree k . We prove the ellipticity of these operators and use this to investigate their kernel, focusing on polynomial solutions. Finally, we will also construct the fundamental solution using the theory of Riesz potentials.
ISSN:0926-2601
1572-929X
DOI:10.1007/s11118-016-9609-3