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Estimates for the full maximal function on graded Lie groups

Duván Cardona

Abstract

On $\mathbb{R}^n,$ a classical result due to Bourgain establishes the restricted weak $(\frac{n}{n-1},1)$ inequality for the full maximal function $M_F^{dσ}$ associated to the spherical averages. In this work we present an extension to Bourgain's result on graded Lie groups for a family of full maximal operators. We formulate this extension using the group Fourier transform of the measures under consideration and the symbols of (positive Rockland operators which are) positive left-invariant hypoelliptic partial differential operators on graded Lie groups.

Estimates for the full maximal function on graded Lie groups

Abstract

On a classical result due to Bourgain establishes the restricted weak inequality for the full maximal function associated to the spherical averages. In this work we present an extension to Bourgain's result on graded Lie groups for a family of full maximal operators. We formulate this extension using the group Fourier transform of the measures under consideration and the symbols of (positive Rockland operators which are) positive left-invariant hypoelliptic partial differential operators on graded Lie groups.
Paper Structure (23 sections, 5 theorems, 132 equations)

This paper contains 23 sections, 5 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Short review of the Euclidean setting
  3. The full maximal function $M_{F}^{d\sigma}$ and admissible measures
  4. Estimates for $M_{F}^{d\sigma}$ on graded Lie groups
  5. Some previous works
  6. Relation with the quantisation on graded Lie groups
  7. Structure of the paper
  8. Preliminaries
  9. Notation
  10. Nilpotent Lie groups
  11. Graded Lie groups $\&$ Rockland operators
  12. Motivating the curvature conditions
  13. The Tauberian type condition
  14. About the differentiability of $t\mapsto \widehat{d\sigma}(t\cdot \pi)$
  15. Proof of the main Theorem \ref{['Maximal:Function:Graded']}
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let $d\sigma$ be a finite Borel measure of compact support on a graded Lie group $G$ of homogeneous dimension $Q.$ Assume that $d\sigma$ is $(Q,Q_0,\varepsilon_0,a)$-admissible (as in Definition Admissible:measure). Let $\mathcal{R}$ be a positive Rockland operator on $G$ of homogeneous degree $\nu> Then $M_{F}^{d\sigma}$ has the following continuity properties:

Theorems & Definitions (14)

  • Definition 1.1: Admissible measures
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Example 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 4.1
  • ...and 4 more