Dimension free bounds for the vector-valued Hardy–Littlewood maximal operator

Abstract

In this article, Fefferman–Stein inequalities in $L^p({\mathbb{R}}^d;{\ell }^q)$ with bounds independent of the dimension $d$ are proved, for all $1<p,q<+\infty$. This result generalizes in a vector-valued setting the famous one by Stein for the standard Hardy–Littlewood maximal operator. We then extend our result by replacing ${\ell }^q$ with an arbitrary UMD Banach lattice. Finally, we prove similar dimensionless inequalities in the setting of the Grushin operators.