Boundedness of the dyadic maximal function on graded Lie groups

Let $1<p\le \mathrm{\infty }$ and let $n\ge 2.$ It was proved independently by C. Calderón, R. Coifman and G. Weiss that the dyadic maximal function \begin{equation*} $\mathcal{M}$^{dσ}Df(x)=\sup{j\in$\mathbb{Z}$}\left|\smallint\limits_{$\mathbb{S}$^{n-1}}f(x-2^jy)dσ(y)\right| \end{equation*} is a bounded operator on $L^p({\mathbb{R}}^n)$ where $d\sigma (y)$ is the surface measure on ${\mathbb{S}}^{n-1}\mathrm{.}$ In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure $d\sigma$ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function ${\mathcal{M}}_D^{d\sigma }$ 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 $\widehat{d\sigma }$ of $d\sigma$ with respect to a fixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of ${\mathcal{M}}_D^{d\sigma }$ on $L^p(G)$ for all $1<p\le \mathrm{\infty }\mathrm{.}$