SPECTRAL MULTIPLIER THEOREMS OF HORMANDER TYPE ON HARDY AND LEBESGUE SPACES

Let X be a space of homogeneous type and let L be an injective, non-negative, self-adjoint operator on L-2(X) such that the semigroup generated by -L fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator F(L), initially defined on H-L(1)(X) boolean AND L2(X), acts as a bounded linear operator on the Hardy space H-L(1)(X) associated with L whenever F is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hormander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators satisfying generalized Gaussian estimates. In this setting our results improve previously known ones.