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

Wolfram Bauer 1  and Lewis A. Coburn 2
  • 1 Institut für Analysis, Leibniz Universität, Welfengarten 1, 30167 Hannover, Germany
  • 2 Department of Mathematics, SUNY at Buffalo, New York 14260, USA

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.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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