Semigroup Kernels, Poisson Bounds, and Holomorphic Functional Calculus

LetLbe the generator of a continuous holomorphic semigroupSwhose action is determined by an integral kernelKon a scale of spacesLp(X; ρ). Under mild geometric assumptions on (X, ρ), we prove that ifLhas a boundedH∞-functional calculus onL2(X; ρ) andKsatisfies bounds typical for the Poisson kernel, thenLhas a boundedH∞-functional calculus onLp(X; ρ) for eachp∈⦠1, ∞⦔. Moreover, if (X, ρ) is of polynomial type andKsatisfies second-order Gaussian bounds, we establish criteria forLto have a bounded Hörmander functional calculus or a bounded Davies–Helffer–Sjöstrand functional calculus.