Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space |
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| Authors: | Qiao Yuying Swanhild Bernstein Sirkka-Liisa John Ryan |
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| Institution: | (1) Department of Mathematics, Hebei Normal University, Shijazhuang, P. R. China;(2) Institute of Mathematics and Physics, Bauhaus University Weimar, D99421 Weimar, Germany;(3) Department of Mathematics, Tampere University of Technology, Tampere, Finland;(4) Department of Mathematics, University of Arkansas, 72701 Fayetteville, AR, USA |
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| 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 forC
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.
Research supported by the National Science Foundation of China (Mathematics Tianyuan Foundation, No A324610) and Hebei Province
(105129)
Research supported by Academy of Finland |
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