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

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 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.