The equality of mixed partial derivatives under weak differentiability conditions

TL;DR

This paper reviews and advances two lesser-known results on the equality of mixed partial derivatives under weak differentiability conditions, with implications for differential geometry and General Relativity.

Contribution

It develops and compares two results on mixed partial derivatives, extending their applicability under weak differentiability assumptions.

Findings

Established conditions for equality at a point and almost everywhere

Connected results to applications in differential geometry and General Relativity

Provided a comprehensive review of Mikusiński and Tolstov's theorems

Abstract

We review and develop two little known results on the equality of mixed partial derivatives which can be considered the best results so far available in their respective domains. The former, due to Mikusi\'nski and his school, deals with equality at a given point, while the latter, due to Tolstov, concerns equality almost everywhere. Applications to differential geometry and General Relativity are commented.