{"title":"尖锐加权对数-索博列夫不等式：平等情况的特征及应用","authors":"Zoltán Balogh, Sebastiano Don, Alexandru Kristály","doi":"10.1090/tran/9163","DOIUrl":null,"url":null,"abstract":"<p>By using optimal mass transport theory, we provide a direct proof to the sharp <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript p\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-log-Sobolev inequality <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis p greater-than-or-equal-to 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(p\\geq 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> involving a log-concave homogeneous weight on an open convex cone <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E subset-of-or-equal-to double-struck upper R Superscript n\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">E\\subseteq \\mathbb R^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript p\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-log-Sobolev inequality. The characterization of the equality cases is new for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than-or-equal-to n\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p\\geq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> even in the unweighted setting and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E equals double-struck upper R Superscript n\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">E=\\mathbb R^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sharp weighted log-Sobolev inequalities: Characterization of equality cases and applications\",\"authors\":\"Zoltán Balogh, Sebastiano Don, Alexandru Kristály\",\"doi\":\"10.1090/tran/9163\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>By using optimal mass transport theory, we provide a direct proof to the sharp <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Superscript p\\\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-log-Sobolev inequality <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis p greater-than-or-equal-to 1 right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(p\\\\geq 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> involving a log-concave homogeneous weight on an open convex cone <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper E subset-of-or-equal-to double-struck upper R Superscript n\\\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">E\\\\subseteq \\\\mathbb R^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Superscript p\\\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-log-Sobolev inequality. The characterization of the equality cases is new for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p greater-than-or-equal-to n\\\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">p\\\\geq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> even in the unweighted setting and <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper E equals double-struck upper R Superscript n\\\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">E=\\\\mathbb R^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.</p>\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-03-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/9163\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9163","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
通过使用最优质量输运理论，我们直接证明了涉及开放凸锥 E ⊆ R n E\subseteq \mathbb R^n 上对数凹同质权的尖锐 L p L^p -log-Sobolev 不等式 ( p ≥ 1 ) (p\geq 1) 。这个证明的好处在于它可以描述实现 L p L^p -log-Sobolev 不等式中相等情形的极值函数。对于 p ≥ n p\geq n，即使是在无权设置和 E = R n E=\mathbb R^n 的情况下，相等情况的特征描述也是新的。作为应用，我们为与汉密尔顿-雅可比方程相关的霍普夫-拉克斯半群提供了一个尖锐的加权超收缩性估计，同时也描述了相等情况。
Sharp weighted log-Sobolev inequalities: Characterization of equality cases and applications
By using optimal mass transport theory, we provide a direct proof to the sharp LpL^p-log-Sobolev inequality (p≥1)(p\geq 1) involving a log-concave homogeneous weight on an open convex cone E⊆RnE\subseteq \mathbb R^n. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the LpL^p-log-Sobolev inequality. The characterization of the equality cases is new for p≥np\geq n even in the unweighted setting and E=RnE=\mathbb R^n. As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.
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