Weighted estimates for a bilinear fractional integral operator and its commutator: A union condition

TL;DR

This paper establishes new weighted boundedness conditions for a bilinear fractional integral operator and its commutator, improving existing results by leveraging the operator's convolution structure and multilinear Muckenhoupt conditions.

Contribution

It introduces a union condition involving multilinear Muckenhoupt-type conditions for weighted boundedness, covering previously unknown cases and enhancing known results.

Findings

New weighted boundedness conditions for $\

Improved results using the convolution structure of the operator

Analysis of commutators and related maximal operators

Abstract

The main theme of this paper is to give sufficient conditions for the weighted boundedness of the bilinear fractional integral operator $\mathsf{BI}_\al$. The proposed condition involves the union of multilinear Muckenhoupt-type conditions. We have achieved new results in an unknown case and remarkably improved other known results by utilizing the hidden convolution nature inside the operator. We also study the effects of the general product commutators on the main operator and the weighted estimates for a related maximal operator that norm-wise dominates the main operator.