Function Theory for Laplace and Dirac-Hodge Operators in Hyperbolic SPace

Details Function Theory for Laplace and Dirac-Hodge Operators in Hyperbolic SPace Function Theory for Laplace and Dirac-Hodge Operators in Hyperbolic SPace

2004-07-12

arxiv.org/abs/math/0407192v1

Mathematics

Analysis of PDEs

23 pages

Scientific paper

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula and a Green's formula for hyperbolic harmonic functions. Using a Cauchy Integral formula we are able to introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions. We introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

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