Boundedness of the dyadic maximal function on graded Lie groups

Let 1 < p≤ ∞ and let n≥ 2. It was proved independently by Calderón, Coifman and Weiss that the dyadic maximal function MdσDf(x)=ⱼ|_Sⁿ⁻¹f(x-2ʲy)dσ(y)| \\[4pt] is a bounded operator on Lᵖ(Rⁿ), where dσ(y) is the surface measure on Sⁿ⁻¹. In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure dσ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function MDdσ using the homogeneous structure of the group. Then, we prove a criterion in terms of the order (at zero and at infinity) of the group Fourier transform dσ of dσ with respect to a fixed Rockland operator R on G that assures the boundedness of MDdσ on Lᵖ(G) for all 1 < p≤ ∞.