$L^{\infty}$ estimates and integrability by compensation in Besov-Morrey spaces and applications

TL;DR

This paper extends $L^{ abla}$ estimates and integrability by compensation results to higher dimensions using Besov-Morrey spaces, enabling new conservation law proofs for elliptic systems with critical regularity.

Contribution

It demonstrates that $L^{ abla}$ estimates hold in arbitrary dimensions within Besov-Morrey spaces, overcoming limitations of previous Morrey space results.

Findings

$L^{ abla}$ estimates valid in any dimension with Besov-Morrey spaces

Existence of conservation laws for elliptic systems with critical regularity

Extension of integrability by compensation techniques to higher dimensions

Abstract

$L^{\infty}$ estimates in the integrability by compensation result of H. Wente fail in dimension larger than two when Sobolev spaces are replaced by the ad-hoc Morrey spaces. However, in this paper we prove that $L^{\infty}$ estimates hold in arbitrary dimension when Morrey spaces are replaced by their Littlewood Paley counterparts: Besov-Morrey spaces. As an application we prove the existence of conservation laws to solution of elliptic systems of the form $-\Delta u= \Omega \cdot \nabla u$ where $\Omega$ is antisymmetric and both $\nabla u$ and $\Omega$ belong to these Besov-Morrey spaces for which the system is critical.