Spectral multipliers on 2-step groups: topological versus homogeneous dimension

Alessio Martini, Detlef Müller

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)
175 Downloads (Pure)

Abstract

Let G be a 2-step stratified group of topological dimension d and homogeneous dimension Q. Let LL be a homogeneous sub-Laplacian on G. By a theorem due to Christ and to Mauceri and Meda, an operator of the form F(L)F(L) is of weak type (1, 1) and bounded on L p (G) for all p ∈ (1, ∞) whenever the multiplier F satisfies a scale-invariant smoothness condition of order s > Q/2. It is known that, for several 2-step groups and sub-Laplacians, the threshold Q/2 in the smoothness condition is not sharp and in many cases it is possible to push it down to d/2. Here we show that, for all 2-step groups and sub-Laplacians, the sharp threshold is strictly less than Q/2, but not less than d/2.
Original languageEnglish
Pages (from-to)680-702
Number of pages23
JournalGeometric and Functional Analysis
Volume26
Issue number2
DOIs
Publication statusPublished - Apr 2016

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