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

• Published in: Geometric and functional analysis Vol. 26; no. 2; pp. 680 - 702
• Main Authors: Martini, Alessio, Müller, Detlef
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2016
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Nilpotent Lie groups; Sub-Laplacians; 43A22; Spectral multipliers; 42B15; Mihlin-Hörmander multipliers; Singular integral operators
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-016-0365-8

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 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.