Sharp Berezin Lipschitz estimates

F. A. Berezin introduced a general "symbol calculus" for linear operators on reproducing kernel Hilbert spaces. For the Segal-Bargmann space H-2( C-n, d mu) of Gaussian square-integrable entire functions on complex n-space, C-n, or for the Bergman spaces A(2)(Omega) of Euclidean volume square-integrable holomorphic functions on bounded domains Omega in C-n, we show here that earlier Lipschitz estimates for Berezin symbols of arbitrary bounded operators are sharp.