Boundedness of the dyadic maximal function on graded Lie groups

Abstract

Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C. Calder\'on, R. Coifman and G. Weiss that the dyadic maximal function \begin{equation*} \mathcal{M}^{d\sigma}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)d\sigma(y)\right| \end{equation*} is a bounded operator on $L^p(\mathbb{R}^n)$ where $d\sigma(y)$ is the surface measure on $\mathbb{S}^{n-1}.$ 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 $1<p\leq \infty.$