Fonctions maximales centrées de Hardy-Littlewood pour les opérateurs de Grushin

Abstract

Let $M_G$ denotes the centered Hardy-Littlewood maximal function associated to the Carnot-Carathéodory distance or to the pseudo-distance associated to the fundamental solution of the Grushin operator on ${\mathbb{R}}_x^n\times {\mathbb{R}}_u$, ${\Delta }_G={\sum }_{i=1}^n\frac{{\partial }^2}{\partial x_i^2}+({\sum }_{i=1}^n{x}_i^2)\frac{{\partial }^2}{\partial u^2}$. We get $L^p$ ($p>1$) dimension free estimates for $M_G$. We prove also that there exists a constant $A>0$ such that $\Vert M_G{\Vert }_{L^1⟶L^{1,\infty }}\le An$, $\forall n$.