Natural annihilators and operators of constant rank over $\mathbb{C}$

TL;DR

This paper investigates the relationship between the nullspaces of constant rank differential operators over complex numbers, showing that equality at the Fourier symbol level implies equivalence of nullspaces modulo polynomials, with applications to Poincaré-type lemmas.

Contribution

It establishes that constant complex rank ensures nullspace equivalence at the operator level when Fourier symbols have the same nullspaces, advancing understanding of annihilators in differential operator complexes.

Findings

Nullspaces of operators with same Fourier symbol nullspaces are equivalent modulo polynomials.

Constant complex rank condition guarantees nullspace equality at the distribution level.

A Poincaré-type lemma is proved for 2D differential operators of constant complex rank.

Abstract

Even if the Fourier symbols of two constant rank differential operators have the same nullspace for each non-trivial phase space variable, the nullspaces of those differential operators might differ by an infinite dimensional space. Under the natural condition of constant rank over $\mathbb{C}$, we establish that the equality of nullspaces on the Fourier symbol level already implies the equality of the nullspaces of the differential operators in $\mathscr{D}'$ modulo polynomials of a fixed degree. In particular, this condition allows to speak of natural annihilators within the framework of complexes of differential operators. As an application, we establish a Poincar\'{e}-type lemma for differential operators of constant complex rank in two dimensions.