Semigroup kernels, Poisson bounds, and holomorphic functional calculus

Abstract

Let L be the generator of a continuous holomorphic semigroup S whose action is determined by an integral kernel K on a scale of spaces Lp(X; ρ). Under mild geometric assumptions on (X, ρ), we prove that if L has a bounded H∞-functional calculus on L2(X; ρ) and K satisfies bounds typical for the Poisson kernel, then L has a bounded H∞-functional calculus on Lp(X; ρ) for each p ∈.