Semigroup kernels, Poisson bounds, and holomorphic functional calculus

Let L be the generator of a continuous holomorphic semigroup S whose action is determined by an integral kernel K on a scare of spaces L(p)(X; rho). Under mild geometric assumptions on (X, rho), we prove that if L has a bounded H-infinity-functional calculus on L(2)(X; rho) and K satisfies bounds typical for the Poisson kernel, then L has a bounded H-infinity-functional calculus on L(p)(X; rho) for each p is an element of [1, infinity]. Moreover, if (X, rho) is of polynomial type and K satisfies second-order Gaussian bounds, we establish criteria for L to have a bounded Hormander functional calculus or a bounded Davies-Helffer-Sjostrand functional calculus. (C) 1996 Academic Press, Inc.