Higher order div-curl type estimates for elliptic linear differential operators on localizable Hardy spaces

TL;DR

This paper develops higher-order div-curl estimates for elliptic linear differential operators on localizable Hardy spaces, extending previous results and introducing new atomic decompositions and inequalities.

Contribution

It introduces higher-order div-curl estimates in local Hardy spaces for elliptic operators, extending prior first-order results and developing new atomic decompositions.

Findings

Extended div-curl estimates to higher orders

Developed a new atomic decomposition for Hardy-Sobolev spaces

Proved a Poincaré-type inequality in this setting

Abstract

In this work, we establish higher-order div-curl type estimates in the sense of Coifman, Lions, Meyer & Semmes, in a local setting for elliptic homogeneous linear differential operators with smooth coefficients acting on localizable Hardy spaces. Our results imply and extend previously known estimates for first-order operators associated with elliptic systems and complexes of vector fields. As tools of independent interest, we develop a new smooth atomic decomposition for localizable Hardy-Sobolev spaces and prove a Poincar\'e-type inequality in this framework.