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Function theory for Laplace and Dirac-Hodge operators in hyperbolic space. (English) Zbl 1135.58013
Summary: 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 for \(C^{1}\) functions and a Green’s formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we 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 and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

58J05 Elliptic equations on manifolds, general theory
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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