Some remarks on vector valued distributions

TL;DR

This paper discusses the regularity properties of fundamental solutions for maximal rank differential operators when their symbols are viewed as vector valued distributions, highlighting that regularity is coordinate-dependent.

Contribution

It reveals that the regularity of fundamental solutions is not invariant under coordinate changes when symbols are considered as vector valued distributions.

Findings

Fundamental solutions may lack regularity despite maximal rank conditions.

Regularity of solutions depends on the choice of local coordinates.

Coordinate invariance of regularity does not hold in this context.

Abstract

We argue that the fundamental solution corresponding to a maximal rank differential operator, where the symbol is regarded as a vector valued distribution, is not necessarily very regular. Thus, for a fundamental solution, the property of being very regular, is not invariant to change of local coordinates.