An existence and uniqueness result about algebras of Schwartz distributions

Abstract

AbstractWe prove that there exists essentially one minimal differential algebra of distributions $$\mathcal A$$ 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 $$\mathcal C_p^{\infty } \subseteq \mathcal A\subseteq \mathcal D' $$ C p ∞ ⊆ A ⊆ D ′ (where $$\mathcal C_p^{\infty }$$ C p ∞ is the set of piecewise smooth functions and $$\mathcal D'$$ D ′ is the set of Schwartz distributions over $$\mathbb R$$ R ). 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 $$\mathcal A$$ A .