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

TL;DR

This paper explores the properties of solutions to Dirac-Hodge and Laplace equations in hyperbolic space, introducing formulas, Hardy spaces, and invariance properties relevant to hyperbolic harmonic and hypermonogenic functions.

Contribution

It develops foundational properties, formulas, and invariance results for hyperbolic harmonic and hypermonogenic functions in hyperbolic space.

Findings

Established Borel-Pompeiu and Green's formulas for hyperbolic harmonic functions.

Introduced Hardy spaces for solutions to the Dirac-Hodge equation.

Demonstrated conformal covariance and invariance properties of hypermonogenic and hyperbolic harmonic functions.

Abstract

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.