On the multiplier problem for the ball on graded Lie groups

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 $p=2.$