Two-weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces

TL;DR

This paper establishes new two-weight norm inequalities for multilinear fractional integrals and maximal functions, providing necessary and sufficient conditions, and extending results to Riesz potentials with product kernels.

Contribution

It introduces novel criteria for two-weight estimates in multilinear fractional integrals and maximal functions, including trace inequalities and Fefferman-Stein type inequalities.

Findings

Derived necessary and sufficient conditions for two-weight inequalities.

Established Fefferman-Stein type inequalities for multilinear operators.

Obtained one-weight criteria for product-type weights.

Abstract

We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.