Boundedness of the dyadic maximal function on graded Lie groups

Abstract Let 1 p and let n 2. It was proved independently by Calderón, Coifman and Weiss that the dyadic maximal function M^dDf (x) =₉|ₒ^₍-₁f (x-2ʲy) d (y) | \\4pt is a bounded operator on Lᵖ (Rⁿ), where d (y) is the surface measure on S^n-1. In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure d with compact support on a graded Lie group G, we associate the corresponding dyadic maximal function MD^d 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 d of d with respect to a fixed Rockland operator R on G that assures the boundedness of MD^d on Lᵖ (G) for all 1 p.