Boundedness of the dyadic maximal function on graded Lie groups

TL;DR

This paper extends the boundedness of dyadic maximal functions from Euclidean spaces to graded Lie groups, providing criteria based on Fourier transform decay for their boundedness on L^p spaces.

Contribution

It introduces a criterion for the boundedness of dyadic maximal functions on graded Lie groups using Fourier transform decay properties.

Findings

Boundedness criterion based on Fourier transform decay

Extension of Euclidean results to graded Lie groups

Applicable for all p in (1, ∞]

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 $1<p\leq \infty.$