{"title":"论费德勒定理中假设的尖锐性","authors":"B. Makarov, A. Podkorytov","doi":"10.1090/spmj/1691","DOIUrl":null,"url":null,"abstract":"<p>The Federer theorem deals with the “massiveness” of the set of critical values for a <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\n <mml:semantics>\n <mml:mi>t</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-smooth map acting from <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript m\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mi>m</mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb R^m</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript n\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mi>n</mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb R^n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>: it claims that the Hausdorff <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-measure of this set is zero for certain <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. If <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to m\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mi>m</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">n\\ge m</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, it has long been known that the assumption of that theorem relating the parameters <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m comma n comma t comma p\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>m</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>p</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">m,n,t,p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is sharp. Here it is shown by an example that this assumption is also sharp for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than m\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>></mml:mo>\n <mml:mi>m</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">n>m</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the sharpness of assumptions in the Federer theorem\",\"authors\":\"B. Makarov, A. Podkorytov\",\"doi\":\"10.1090/spmj/1691\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Federer theorem deals with the “massiveness” of the set of critical values for a <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"t\\\">\\n <mml:semantics>\\n <mml:mi>t</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">t</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-smooth map acting from <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R Superscript m\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mi>m</mml:mi>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb R^m</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> to <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R Superscript n\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mi>n</mml:mi>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb R^n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>: it claims that the Hausdorff <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\">\\n <mml:semantics>\\n <mml:mi>p</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-measure of this set is zero for certain <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\">\\n <mml:semantics>\\n <mml:mi>p</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. If <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n greater-than-or-equal-to m\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>n</mml:mi>\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mi>m</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">n\\\\ge m</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, it has long been known that the assumption of that theorem relating the parameters <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"m comma n comma t comma p\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>m</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>n</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>p</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">m,n,t,p</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is sharp. 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引用次数: 0
摘要
费德勒定理处理的是从R m \mathbb R^m到R n \mathbb R^n的光滑映射的临界值集合的“海量性”:它声称该集合的Hausdorff p p -测度在特定的p p下为零。如果n≥m n\ge m，我们早就知道关于参数m,n,t,p m,n,t,p的定理的假设是尖锐的。这里的一个例子表明，这个假设对于n b>00 m和n bb11m也是尖锐的。
On the sharpness of assumptions in the Federer theorem
The Federer theorem deals with the “massiveness” of the set of critical values for a tt-smooth map acting from Rm\mathbb R^m to Rn\mathbb R^n: it claims that the Hausdorff pp-measure of this set is zero for certain pp. If n≥mn\ge m, it has long been known that the assumption of that theorem relating the parameters m,n,t,pm,n,t,p is sharp. Here it is shown by an example that this assumption is also sharp for n>mn>m.
期刊介绍：
This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.
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