A Sparse Estimate for Multisublinear Forms Involving Vector-valued Maximal Functions

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 ℓ^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 developed by Cruz-Uribe and Martell in Limited range multilinear extrapolation with applications to the bilinear Hilbert transform, preprint arXiv:1704.06833 (2017). As an example, vector-valued multilinear weighted inequalities for bilinear Hilbert transforms are deduced from the scalar sparse domination theorem of Domination of multilinear singular integrals by positive sparse forms, preprint arXiv:1603.05317.