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Accessible Unlicensed Requires Authentication Published by De Gruyter March 31, 2015

Heat flow, weighted Bergman spaces, and real analytic Lipschitz approximation

Wolfram Bauer and Lewis A. Coburn

Abstract

We show that, for f any uniformly continuous (UC) complex-valued function on real Euclidean n-space ℝn, the heat flow f˜(t) is Lipschitz for all t > 0 and f˜(t) converges uniformly to f as t → 0. Analogously, let Ω be any irreducible bounded symmetric (Cartan) domain in complex n-space ℂn and consider the Bergman metric β(·, ·) on Ω. For f any β-uniformly continuous function on Ω, we show that there is a Berezin–Harish-Chandra flow of real analytic functions Bλf which is β-Lipschitz for each λ ≥ p (p, the genus of Ω) and Bλf converges uniformly to f as λ → ∞. For a certain subspace of UC we obtain stronger approximation results and we study the asymptotic behaviour of the Lipschitz constants.

Funding source: DFG (Deutsche Forschungsgemeinschaft)

Award Identifier / Grant number: Emmy-Noether scholarship

Received: 2013-12-2
Published Online: 2015-3-31
Published in Print: 2015-6-1

© 2015 by De Gruyter