{"title":"An existence and uniqueness result about algebras of Schwartz distributions","authors":"Nuno Costa Dias, Cristina Jorge, João Nuno Prata","doi":"10.1007/s00605-023-01917-z","DOIUrl":null,"url":null,"abstract":"Abstract We prove that there exists essentially one minimal differential algebra of distributions $$\\mathcal A$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>A</mml:mi> </mml:math> , satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l’impossibilité de la multiplication des distributions, 1954], and such that $$\\mathcal C_p^{\\infty } \\subseteq \\mathcal A\\subseteq \\mathcal D' $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:mo>⊆</mml:mo> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> (where $$\\mathcal C_p^{\\infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:math> is the set of piecewise smooth functions and $$\\mathcal D'$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> is the set of Schwartz distributions over $$\\mathbb R$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>R</mml:mi> </mml:math> ). 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 $$\\mathcal A$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>A</mml:mi> </mml:math> .","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-023-01917-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
Abstract We 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 $$\mathcal C_p^{\infty } \subseteq \mathcal A\subseteq \mathcal D' $$ Cp∞⊆A⊆D′ (where $$\mathcal C_p^{\infty }$$ Cp∞ 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 .
摘要证明了存在一个极小的分布微分代数 $$\mathcal A$$ A,满足Schwartz不可能结果中的所有性质[L]。Schwartz, Sur l ' impossible it de la multiplication des distribution(1954),等等 $$\mathcal C_p^{\infty } \subseteq \mathcal A\subseteq \mathcal D' $$ C p∞(其中 $$\mathcal C_p^{\infty }$$ C p∞是分段光滑函数和的集合 $$\mathcal D'$$ D '是Schwartz分布的集合 $$\mathbb R$$ R)。这个代数被赋予了分布的乘法积,它是在[N.C.]中定义的乘积的推广Dias, J.N.Prata,一类具有分布系数的常微分方程的乘积[j]。如果代数不是极小的,但满足上述条件,在反微分和光滑函数的对偶积下是封闭的,并且分布积在零处连续,那么它必然是一个扩展 $$\mathcal A$$ 选A。