Swanhild Bernstein — Page 2

arXiv2004

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

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.

math.AP↗

arXiv1998

Monotonicity principles for singular integral equations in Clifford analysis

Monotonicity principles can be used to get informations about nonlinear singular integral equations. These results are based on a theorem of Browder and Minty. We consider a family of monotone singular integral operators and associated nonlinear singular integral equations with Nemickii operator in Clifford analysis which has an application to the nonlinear magnetic field equation in the 3-dimensional Euclidean space.

math.CV↗