Oscillating spectral multipliers on groups of Heisenberg type

Roberto Bramati,John Green,Paolo Ciatti,James Wright

doi:10.4171/rmi/1302

Abstract

We establish endpoint estimates for a class of oscillating spectral multipliers on Lie groups of Heisenberg type. The analysis follows an earlier argument due to the second and fourth author \[Springer INdAM Ser., vol. 45 (2021)], but requires the detailed analysis of the wave equation on these groups due to Müller and Seeger \[Anal. PDE 8 (2015)]. We highlight and develop the connection between sharp bounds for oscillating spectral multipliers and the problem of determining the minimal amount of smoothness required for Mihlin–Hörmander multipliers, a problem that has been solved for groups of Heisenberg type but remains open for other groups.

Highlights

Let G be a connected Lie group and let X1, X2, . . . , Xn be left invariant vector fields on G which satisfy Hormander’s condition; that is, they generate, together with the iterative commutators, the tangent space of G at every point.The sublaplacian L = − n j=1Xj2 is a nonnegative, second order hypoelliptic operator which√is essentially self-adjoint on L2(μ) where μ is the right Haar measure on G
In this paper we will consider a general framework of spectral multipliers which contains oscillating examples of the form mθ,β (λ) eiλθ λθβ/2 χ(λ) for any θ, β ≥ 0
We review the definition of Lie groups of Heisenberg type and recall the key results from [13] where Muller and Seeger give a detailed analysis of the wave equation in this setting, including the introduction of a local, isotropic Hardy space h1iso(G) which is compatible with the underlying group structure

Summary

Introduction

Let G be a connected Lie group and let X1, X2, . . . , Xn be left invariant vector fields on G which satisfy Hormander’s condition; that is, they generate, together with the iterative commutators, the tangent space of G at every point (a special case is a Lie group of Heisenberg type; see the section for precise definitions). In [3], the lower bound d/2 ≤ s−(L) was established in great generality, including sublaplacians L on any stratified Lie group of arbitrary step. The same conclusions were obtained in [10] in less generality; for sublaplacians on any step 2 stratified group, with less robust methods Another threshold exponent s+(L) was introduced in [10] which will be useful for us. We review the definition of Lie groups of Heisenberg type and recall the key results from [13] where Muller and Seeger give a detailed analysis of the wave equation in this setting, including the introduction of a local, isotropic Hardy space h1iso(G) which is compatible with the underlying group structure. Where x ∈ g1 and u ∈ g2 and Jx, x′ denotes a vector in g2 with components JUi x, x′

Consider the positive sublaplacian

To treat the remaining part

Next we will show that