Fourier multipliers on graded Lie groups

TL;DR

This paper investigates Fourier multipliers on graded nilpotent Lie groups, establishing $L^p$-boundedness under Hörmander-type conditions and introducing refined Sobolev space criteria using difference operators and Rockland operators.

Contribution

It extends multiplier theory to graded Lie groups by formulating Hörmander conditions with difference operators and Sobolev spaces, providing new criteria for boundedness.

Findings

Hörmander conditions imply $L^p$-boundedness on graded Lie groups

Introduction of Sobolev space conditions on the dual group

Use of Rockland operators and difference operators in multiplier analysis

Abstract

In this paper we study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that H\"ormander type conditions on the Fourier multipliers imply $L^p$-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper.