Fonctions maximales centr\'ees de Hardy-Littlewood pour les op\'erateurs de Grushin

TL;DR

This paper establishes dimension-free $L^p$ bounds and weak-type estimates for the centered Hardy-Littlewood maximal function associated with the Grushin operator on $ ^n imes $, extending harmonic analysis tools to this subelliptic setting.

Contribution

It provides the first dimension-free $L^p$ estimates and weak-type bounds for the maximal function related to the Grushin operator, a key step in analysis on subelliptic spaces.

Findings

Dimension-free $L^p$ bounds for $p > 1$

Weak-type $(1,1)$) estimate grows linearly with dimension

Extension of harmonic analysis techniques to Grushin-type operators

Abstract

Let $M_G$ denotes the centered Hardy-Littlewood maximal function associated to the Carnot-Carath\'eodory distance or to the pseudo-distance associated to the fundamental solution of the Grushin operator on $\R_x^n \times \R_u$, $\Delta_G = \sum_{i = 1}^n \frac{\partial^2}{\partial x_i^2} + (\sum_{i = 1}^n x_i^2) \frac{\partial^2}{\partial u^2}$. We get $L^p$ ($p > 1$) dimension free estimates for $M_G$. We prove also that there exists a constant $A > 0$ such that $\| M_G \|_{L^1 \longrightarrow L^{1, \infty}} \leq A n$, $\forall n$.