Differential operators on Schwartz distributions. Jet formalism

TL;DR

This paper explores the algebraic formalism of differential operators on Schwartz distributions within the jet formalism, extending the conventional definitions to encompass a broader class of operators on distributions over smooth manifolds.

Contribution

It introduces an algebraic approach to differential operators on Schwartz distributions, generalizing the classical transpose-based definition to a more comprehensive framework.

Findings

Extended the class of differential operators on Schwartz distributions.

Provided a formal algebraic framework using jet formalism.

Applied the theory to distributions on sections of vector bundles.

Abstract

Differential operators on Schwartz distributions conventionally are defined as the transpose of differential operators on functions with compact support. They do not exhaust all differential operators. We follow algebraic formalism of differential operators on modules over commutative rings. In a general setting, Schwartz distributions on sections with compact support of vector bundles over an arbitrary smooth manifold are considered.