On a variant of Tartar's first commutation lemma

TL;DR

This paper presents a modified version of Tartar's first commutation lemma that applies to multiplier operators with symbols not confined to a codimension-one manifold, expanding the lemma's applicability.

Contribution

It introduces a novel variant of Tartar's lemma accommodating more general symbol conditions beyond the traditional manifold setting.

Findings

Extended the lemma to broader classes of multiplier operators.

Provided rigorous proof of the new variant.

Potential applications in harmonic analysis and PDEs.

Abstract

We prove a variant of Tartar's first commutation lemma involving multiplier operators with symbols not necessarily defined on a manifold of codimension one.