H-distributions - an extension of the H-measures

TL;DR

This paper introduces H-distributions, extending H-measures to the L^p setting, and applies them to localise properties and reprove a div-curl lemma variant, advancing analysis tools for PDEs.

Contribution

It extends H-measures to H-distributions in L^p spaces and applies them to localisation principles and div-curl lemmas.

Findings

H-distributions generalize H-measures to L^p spaces

Established an L^p localisation principle

Reproved an L^p-L^q div-curl lemma

Abstract

We use the continuity of Fourier multiplier operators on $L^p$ to introduce the $H$-distributions --- an extension of $H$-measures in the $L^p$ framework. We apply the $H$-distributions to obtain an $L^p$ version of the localisation principle, and reprove the $L^p$---$L^q$ variant of the Murat--Tartar div-curl lemma.