Local linear dependence of linear partial differential operators

TL;DR

This paper establishes that finite sets of linear partial differential operators with continuous coefficients are linearly dependent if and only if they are locally linearly dependent, leading to a weak nullstellensatz for these operators.

Contribution

It proves the equivalence of global and local linear dependence for finite sets of linear PDE operators with continuous coefficients.

Findings

Global linear dependence iff local linear dependence

Reflexive closure equals linear span for finite sets

Weak nullstellensatz for linear PDE operators

Abstract

We show that any finite set of linear partial differential operators with continuous coefficients is linearly dependent if and only if it is locally linearly dependent. It follows that the reflexive closure of any finite set of such operators is equal to its linear span. The last statement can be rephrased as a weak nullstellensatz for linear partial differential operators.