Weighted Inequalities for Multilinear Potential Operators and their Commutators

Abstract

We prove weighted strong inequalities for the multilinear potential operator \({\cal T}_{\phi}\) and its commutator, where the kernel ϕ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type estimates. Moreover we prove weighted weak type inequalities for the multilinear maximal operator \(\mathcal{M}_{\varphi,L^{B}}\) associated to an essentially nondecreasing function φ and to the Orlicz space LB for a given Young function B. This result allows us to obtain a weighted weak type inequality for the operator \({\cal T}_{\phi}\).