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.
|Number of pages||23|
|Journal||Geometric and Functional Analysis|
|Publication status||Published - Apr 2016|