On the multiplier problem for the ball on graded Lie groups

TL;DR

This paper explores a non-commutative analogue of the classical multiplier problem for the ball, focusing on differential operators on graded Lie groups and establishing boundedness conditions for their multipliers.

Contribution

It introduces a non-commutative version of the Fefferman multiplier problem for graded Lie groups and characterizes the boundedness of associated multipliers.

Findings

Multipliers are bounded on L^p(G) only when p=2.

The study extends classical multiplier problems to non-commutative settings.

Provides conditions under which differential operator multipliers are bounded.

Abstract

In this note, we consider a non-commutative analogy of the classical Fefferman multiplier problem for the ball. More precisely, if $\chi$ is the characteristic function of the unit interval $I=[0,1],$ we investigate a family of differential operators $\mathcal{R}$ on a graded Lie group $G,$ for which the multipliers $\chi(\mathcal{R})$ are bounded on $L^p(G),$ if and only if $p=2.$