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arXiv · 2309.14444

An existence and uniqueness result about algebras of Schwartz distributions

Nuno Costa Dias·Cristina Jorge·Joao Nuno Prata
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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'impossibilité 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 $\A$.

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Nuno Costa Dias, Cristina Jorge, Joao Nuno Prata. 2023-09-25. An existence and uniqueness result about algebras of Schwartz distributions. https://doi.org/10.1007/s00605-023-01917-z

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