Bilinear Fractional Integral Operators

TL;DR

This paper characterizes the boundedness of bilinear fractional integral operators involving matrix coefficients across various function spaces, extending previous work by Kenig and Stein.

Contribution

It provides a complete parameter characterization for the boundedness of these operators in more general settings than previously studied.

Findings

Complete parameter characterization for boundedness.

Extension of Kenig and Stein's work to more general settings.

Conditions for boundedness from product spaces to target Lebesgue spaces.

Abstract

We study the bilinear fractional integral considered by Kenig and Stein, where linear combinations of variables with matrix coefficients are involved. Under more general settings, we give a complete characterization of the corresponding parameters for which the bilinear fractional integral is bounded from $L^{p_1}(\mathbb R^{n_1}) \times L^{p_2}(\mathbb R^{n_2})$ to $L^q(\mathbb R^m)$.