Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth

TL;DR

This paper investigates the relationship between convolution kernels and spectral multipliers for sub-Laplacians on groups of polynomial growth, establishing a converse result and exploring kernel integrability implications.

Contribution

It proves a converse implication for the Schwartz class of spectral multipliers on certain groups, including solvable noncompact groups of polynomial growth.

Findings

Convolution kernel Schwartz class implies spectral multiplier Schwartz class.

Established a converse for a class of groups including solvable noncompact groups.

Discussed whether kernel integrability implies continuity of spectral multipliers.

Abstract

Let $\mathcal{L}$ be a sub-Laplacian on a connected Lie group $G$ of polynomial growth. It is well known that, if $F : \mathbb{R} \to \mathbb{C}$ is in the Schwartz class $\mathcal{S}(\mathbb{R})$, then the convolution kernel $\mathcal{K}_{F(\mathcal{L})}$ of the operator $F(\mathcal{L})$ is in the Schwartz class $\mathcal{S}(G)$. Here we prove a sort of converse implication for a class of groups $G$ including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of $\mathcal{K}_{F(\mathcal{L})}$ implies continuity of $F$.