An existence and uniqueness result about algebras of Schwartz distributions

TL;DR

This paper establishes the uniqueness of a minimal differential algebra of distributions that extends piecewise smooth functions and Schwartz distributions, satisfying Schwartz's impossibility conditions and generalizing previous distribution products.

Contribution

It proves the existence and essential uniqueness of a minimal algebra of distributions with a multiplicative product, extending prior work and satisfying key properties.

Findings

Existence of a unique minimal algebra of distributions.

Extension of the product defined in prior research.

Any algebra satisfying certain conditions extends this minimal algebra.

Abstract

We prove that there exists essentially one {\it minimal} differential algebra of distributions $\A$, satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l'impossibilit\'e de la multiplication des distributions, 1954], and such that $\C_p^{\infty} \subseteq \A \subseteq \DO' $ (where $\C_p^{\infty}$ is the set of piecewise smooth functions and $\DO'$ is the set of Schwartz distributions over $\RE$). This algebra is endowed with a multiplicative product of distributions, which is a generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients, 2009]. If the algebra is not minimal, but satisfies the previous conditions, is closed under anti-differentiation and the dual product by smooth functions, and the distributional product is continuous at zero then it is necessarily an extension of $\A$.