Spectral multiplier theorems of Hörmander type on Hardy and Lebesgue spaces

2012/09/03 by Kunstmann, Peer Christian, Uhl, Matthias · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1209.0358

Abstract

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 H¹_L(X)∩ L²(X), acts as a bounded linear operator on the Hardy space H¹_L(X) 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 which the required differentiability order is relaxed compared to all known spectral multiplier results.

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