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

On the multiplier problem for the ball on graded Lie groups