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

In this article, Fefferman-Stein inequalities in $L^p({\mathbb{R}}^d;{\mathrm{\ell }}^q)$ 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 ${\mathrm{\ell }}^q$ with an arbitrary UMD Banach lattice. Finally,we prove similar dimensionless inequalities in the setting of the Grushinoperators.