Fractional integral operator for $L^1$ vector fields and its applications

TL;DR

This paper investigates fractional integral operators for vector fields in weighted L^1 spaces, providing new proofs of classical inequalities and establishing novel divergence-curl inequalities using advanced harmonic analysis techniques.

Contribution

It introduces new methods to analyze fractional integral operators in weighted L^1 spaces and derives novel divergence-curl inequalities for vector fields.

Findings

New proof for Caffarelli-Kohn-Nirenberg inequalities

Establishment of new divergence-curl inequalities

Application of Stein-Weiss inequalities in vector field analysis

Abstract

This paper studies fractional integral operator for vector fields in weighted $L^1$. Using the estimates on fractional integral operator and Stein-Weiss inequalities, we can give a new proof for a class of Caffarelli-Kohn-Nirenberg inequalities and establish new $\divg$-$\curl$ inequalities for vector fields.