Let G be a 2-step stratified group of topological dimension d and homogeneous dimension Q. Let L 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) is of weak type (1, 1) and bounded on Lp(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.

Spectral multipliers on 2-step groups: topological versus homogeneous dimension / Martini, A.; Muller, D.. - In: GEOMETRIC AND FUNCTIONAL ANALYSIS. - ISSN 1016-443X. - STAMPA. - 26:2(2016), pp. 680-702. [10.1007/s00039-016-0365-8]

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

Martini A.;
2016

Abstract

Let G be a 2-step stratified group of topological dimension d and homogeneous dimension Q. Let L 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) is of weak type (1, 1) and bounded on Lp(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.
File in questo prodotto:
File Dimensione Formato  
Martini-Müller2016_Article_SpectralMultipliersOn2-stepGro.pdf

accesso aperto

Tipologia: 2a Post-print versione editoriale / Version of Record
Licenza: Creative commons
Dimensione 615.9 kB
Formato Adobe PDF
615.9 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2949502