# Journal of Operator Theory

Volume 73, Issue 1, Winter 2015  pp. 27-69.

Spectral multiplier theorems of Hormander type on Hardy and Lebesgue spaces

Authors:  Peer Christian Kunstmann (1) and Matthias Uhl (2)
Author institution:(1) Department of Mathematics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany
(2) Department of Mathematics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany

Summary: 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 $H^1_L(X)\cap L^2(X)$, acts as a bounded linear operator on the Hardy space $H^1_L(X)$ 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.

DOI: http://dx.doi.org/10.7900/jot.2013aug29.2038
Keywords: spectral multiplier theorems, Hardy spaces, non-negative self-adjoint operators, Davies--Gaffney estimates, spaces of homogeneous type