A sparse estimate for multisublinear forms involving vector-valued maximal functions

TL;DR

This paper establishes a sparse bound for vector-valued maximal functions and demonstrates how these bounds extend to weighted inequalities for multisublinear operators, including bilinear Hilbert transforms.

Contribution

It introduces a sparse bound for vector-valued maximal functions and shows the preservation of sparse bounds through $\,\ell^r$-valued extensions, enabling new weighted inequalities.

Findings

Sparse bounds for vector-valued maximal functions are proven.

Sparse bounds are preserved under $\,\ell^r$-valued extensions.

Vector-valued weighted inequalities for bilinear Hilbert transforms are derived.

Abstract

We prove a sparse bound for the $m$-sublinear form associated to vector-valued maximal functions of Fefferman-Stein type. As a consequence, we show that the sparse bounds of multisublinear operators are preserved via $\ell^r$-valued extension. This observation is in turn used to deduce vector-valued, multilinear weighted norm inequalities for multisublinear operators obeying sparse bounds, which are out of reach for the extrapolation theory recently developed by Cruz-Uribe and Martell. As an example, vector-valued multilinear weighted inequalities for bilinear Hilbert transforms are deduced from the scalar sparse domination theorem of the authors.