Properties and applications of partial multiple weights for fractional integrals

TL;DR

This paper introduces partial multiple weights for fractional integrals, establishing new extrapolation theorems, weighted estimates, and inequalities, and characterizing commutators, thus advancing the understanding of weighted fractional integral operators.

Contribution

It develops the theory of partial Muckenhoupt weights, proving new extrapolation theorems and inequalities, and addresses an open question on commutators of fractional integrals.

Findings

Established Rubio de Francia extrapolation theorem for partial Muckenhoupt classes.

Derived weighted estimates for fractional integrals with partial weights.

Characterized commutators of fractional integrals, answering an open question.

Abstract

In this paper, through the introduction of partial multiple weights, we firstly study the related Rubio de Francia extrapolation theorem within the framework of partial Muckenhoupt classes and further obtain the corresponding extrapolation theorem for two types of off-diagonal estimates. Secondly, we establish some weighted estimates for fractional integrals associated with partial Muckenhoupt weights. As applications, several basic inequalities (including the Fefferman-Phong inequality, the degenerate Poincar\'{e} inequality and the Caffarelli-Kohn-Nirenberg inequality) related to partial Muckenhoupt weights are derived. Meanwhile, our results can give the characterization of the commutators of fractional integrals, which yields a partial answer to an open question proposed by D. Cruz-Uribe in the paper [D. Cruz-Uribe, Two weight inequalities for fractional integral operators and commutators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, 25-85].