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$L^2$-Maximal functions on graded Lie groups

Duván Cardona

Abstract

Bourgain in his seminal paper [2] about the analysis of maximal functions associated to convex bodies, has estimated in a sharp way the $L^2$-operator norm of the maximal function associated to a kernel $K\in L^1,$ with differentiable Fourier transform $\widehat{K}.$ We formulate the extension to Bourgain's $L^2$-estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the $L^p$-boundedness of maximal functions on graded Lie groups.

$L^2$-Maximal functions on graded Lie groups

Abstract

Bourgain in his seminal paper [2] about the analysis of maximal functions associated to convex bodies, has estimated in a sharp way the -operator norm of the maximal function associated to a kernel with differentiable Fourier transform We formulate the extension to Bourgain's -estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the -boundedness of maximal functions on graded Lie groups.
Paper Structure (9 sections, 3 theorems, 60 equations)

This paper contains 9 sections, 3 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. $L^2$-estimates for maximal functions on $\mathbb{R}^n$
  3. $L^2$-estimates for maximal functions on graded Lie groups
  4. Fourier analysis on nilpotent Lie groups and Rockland operators
  5. Nilpotent Lie groups
  6. The unitary dual $\widehat{G}$
  7. Plancherel theorem on nilpotent Lie groups
  8. Rockland operators on graded Lie groups
  9. Estimation of maximal functions using the Fourier transform

Key Result

Theorem 1.1

Let $K\in L^1(\mathbb{R}^n)$ be such that its Fourier transform $\widehat{K}$ is differentiable. Define the following quantities Then, we have the following estimate on the maximal operator associated to $K,$ for every $f\in \mathscr{S}(\mathbb{R}^n),$ where

Theorems & Definitions (10)

  • Theorem 1.1: Bourgain Bourgain1986
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 3.1
  • proof : Proof of Theorem \ref{['main:theorem']}
  • Remark 3.2