| I study the structure of the Berezin symbol calculus of general operators on Bergman reproducing kernel Hilbert spaces, and investigate how the Bergman geometry of the underlying domains affects the boundary behavior of the functions in the range of the Berezin transform by establishing certain regularity estimates for Berezin symbols on various domains in Cn . It is shown by Coburn [14], [15] that the Berezin symbols X˜ of general bounded operators must satisfy certain severe Lipschitz growth limitations which were not previously known. Moreover, a strengthened version---pointed out by Xia---that X˜ and its partial derivatives of all orders are bounded, holds on the Segal-Bargmann space on Cn . For the unit ball, I show that the analog of Xia's result remains in force for the algebra of invariant differential operators (generated by the Laplace-Beltrami operator). This result appeared in [28]. For general bounded domains in Cn , the infinitesimal version of Coburn's estimates is obtained in [16] as the "sharp directional derivative estimates". Finally, I extend the above results to obtain Bloch-like estimates for the higher order derivatives of general Berezin symbols. This analysis depends upon a careful study of Burbea's mth order Bergman metric. |
