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

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