# 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

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