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

## TL;DR

This paper establishes dimension-free bounds for vector-valued Hardy-Littlewood maximal operators in various Banach space settings, extending classical results and applying to Grushin operators, thus advancing the understanding of maximal inequalities in high-dimensional analysis.

## Contribution

It generalizes Fefferman-Stein inequalities to vector-valued functions in UMD Banach lattices with bounds independent of dimension, extending classical scalar results.

## Key findings

- Dimension-free bounds for vector-valued maximal operators in $L^p(
^d;ell^q)$
- Extension of results to UMD Banach lattices
- Dimensionless inequalities for Grushin operators

## Abstract

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