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Boundedness of the dyadic maximal function on graded Lie groups

Duván Cardona, Julio Delgado, Michael Ruzhansky

Abstract

Let $1<p\leq \infty$ and let $n\geq 2.$ 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 $dσ(y)$ 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 $dσ$ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function $\mathcal{M}_D^{dσ}$ 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 $\widehat{dσ}$ of $dσ$ with respect to a fixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of $\mathcal{M}_D^{dσ}$ on $L^p(G)$ for all $1<p\leq \infty.$

Boundedness of the dyadic maximal function on graded Lie groups

Abstract

Let and let 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 where is the surface measure on In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure with compact support on a graded Lie group we associate the corresponding dyadic maximal function 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 of with respect to a fixed Rockland operator on that assures the boundedness of on for all
Paper Structure (15 sections, 5 theorems, 127 equations)

This paper contains 15 sections, 5 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. The main result
  4. Fourier analysis on graded groups
  5. Homogeneous and graded Lie groups
  6. Fourier analysis on nilpotent Lie groups
  7. Homogeneous linear operators and Rockland operators
  8. Boundedness of the dyadic maximal function
  9. The key lemma
  10. Step 1.
  11. Step 2.
  12. The probabilistic argument
  13. Proof of the main theorem
  14. Conflict of interests statement.
  15. Data Availability Statements.

Key Result

Theorem 1.1

Let $d\sigma$ be a finite Borel measure of compact support on a graded Lie group $G.$ Let $\mathcal{R}$ be a positive Rockland operator on $G$ of homogeneous degree $\nu>0.$ Assume that for some $a>0$ the Fourier transform of $d\sigma$ satisfies the growth estimate Then, the dyadic maximal function $\mathcal{M}^{d\sigma}_D:L^p(G)\rightarrow L^p(G)$ can be extended to a bounded operator for all $1

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 26 more