SPECTRAL MULTIPLIER THEOREMS OF HÖRMANDER 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 L2(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 ${\mathrm{H}}_{\mathrm{L}}^{1}\left(\mathrm{X}\right)\cap {\mathrm{L}}^{2}\left(\mathrm{X}\right)$, acts as a bounded linear operator on the Hardy space ${\mathrm{H}}_{\mathrm{L}}^{1}\left(\mathrm{X}\right)$ associated with L whenever F is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hörmander 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.