On the endomorphisms of some sheaves of functions

TL;DR

This paper characterizes the sheaf of differential operators of a certain order and regularity on a smooth manifold as a sheaf of morphisms between sheaves of differentiable functions, linking operators to sheaf homs.

Contribution

It establishes an explicit description of the sheaf of differential operators in terms of sheaf homomorphisms between sheaves of differentiable functions with different regularities.

Findings

Sheaf of differential operators can be expressed via sheaf homs.

Provides a sheaf-theoretic description of differential operators.

Connects differential operators to morphisms of function sheaves.

Abstract

Given a $C^\infty$ real manifold $X$ and $\mathcal{C}^m_X$ its sheaf of $m$-times differentiable real-valued functions, we prove that the sheaf $\mathcal{D}^{m, r}_X$ of differential operators of order $\leq m$ with coefficient functions of class $C^r$ can be obtained in terms of the sheaf $\mathcal{H}om_{\mathbb{R}_X}(\mathcal{C}^m_X, \mathcal{C}^r_X)$ of morphisms of $\mathcal{C}^m_X$ into $\mathcal{C}^r_X$. The superscripts $m$ and $r$ are integers.