On Fuglede's flux extensions and the point wise definition of linear partial differential operators

Abstract

In this work we provide a survey of Fuglede's flux extensions of first order partial differential operators, a concept largely forgotten today. A long the way we also survey the classical weak and strong extensions of PDE operators and the works of Friedrichs and H\"ormander. We give several applications of this theory showing its usefulness, as well as connecting it to more recent developments in connection to various sharp versions of the divergence theorem. In particular, we use it to prove a generalization of Morera's theorem valid for general first order operators. Using this theory we also prove a new local limit formula for the maximal extension of a first order operator. We initiate a study of this limit and connect it to the wave cone of the operator, a concept that first arose in the theory of compensated compactness. Hopefully, this will contribute to a rival of Fuglede's beautiful ideas.