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

Abstract

Abstract We show that, for f any uniformly continuous (UC) complex-valued function on real Euclidean n-space ℝ n , 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 ℂ n 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 B λ f which is β-Lipschitz for each λ ≥ p (p, the genus of Ω) and B λ f 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.