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 R-n, the heat flow (f) over tilde ((t)) is Lipschitz for all t > 0 and (f) over tilde ((t)) converges uniformly to f as t -> 0. Analogously, let Omega be any irreducible bounded symmetric (Cartan) domain in complex n-space C-n and consider the Bergman metric beta(. , .) on Omega. For f any beta-uniformly continuous function on Omega, we show that there is a Berezin-Harish-Chandra flow of real analytic functions B(lambda)f which is beta-Lipschitz for each lambda >= p (p, the genus of Omega) and B(lambda)f converges uniformly to f as lambda -> infinity. For a certain subspace of UC we obtain stronger approximation results and we study the asymptotic behaviour of the Lipschitz constants.