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

$L^{\infty}$ 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 $-Δu= Ω\cdot \nabla u$ where $Ω$ is antisymmetric and both $\nabla u$ and $Ω$ belong to these Besov-Morrey spaces for which the system is critical.

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