{"title":"在一个弦弧曲线上的𝐿^{𝑝}范数中的Hölder类","authors":"T. Alexeeva, N. Shirokov","doi":"10.1090/spmj/1769","DOIUrl":null,"url":null,"abstract":"<p>The Hölder classes <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p Superscript alpha Baseline left-parenthesis upper L right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi>L</mml:mi>\n <mml:mi>p</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>α<!-- α --></mml:mi>\n </mml:mrow>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L_p^{\\alpha } (L)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p Baseline left-parenthesis upper L right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>p</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L_p(L)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> norm on a <italic>chord-arc</italic> curve <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R cubed\">\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:mn>3</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}^3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript p Baseline left-parenthesis upper L right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>L</mml:mi>\n <mml:mi>p</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L^p(L)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> norm, the direct theorem is proved for a certain subclass of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p Superscript alpha Baseline left-parenthesis upper L right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi>L</mml:mi>\n <mml:mi>p</mml:mi>\n <mml:mi>α<!-- α --></mml:mi>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L^\\alpha _p(L)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and the inverse theorem covers the entire Hölder class.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Hölder classes in the 𝐿^{𝑝} norm on a chord-arc curve in ℝ³\",\"authors\":\"T. Alexeeva, N. Shirokov\",\"doi\":\"10.1090/spmj/1769\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Hölder classes <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript p Superscript alpha Baseline left-parenthesis upper L right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msubsup>\\n <mml:mi>L</mml:mi>\\n <mml:mi>p</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>α<!-- α --></mml:mi>\\n </mml:mrow>\\n </mml:msubsup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>L</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L_p^{\\\\alpha } (L)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript p Baseline left-parenthesis upper L right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>L</mml:mi>\\n <mml:mi>p</mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>L</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L_p(L)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> norm on a <italic>chord-arc</italic> curve <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L\\\">\\n <mml:semantics>\\n <mml:mi>L</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R cubed\\\">\\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:mn>3</mml:mn>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}^3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Superscript p Baseline left-parenthesis upper L right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msup>\\n <mml:mi>L</mml:mi>\\n <mml:mi>p</mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>L</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L^p(L)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> norm, the direct theorem is proved for a certain subclass of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript p Superscript alpha Baseline left-parenthesis upper L right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msubsup>\\n <mml:mi>L</mml:mi>\\n <mml:mi>p</mml:mi>\\n <mml:mi>α<!-- α --></mml:mi>\\n </mml:msubsup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>L</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L^\\\\alpha _p(L)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and the inverse theorem covers the entire Hölder class.</p>\",\"PeriodicalId\":51162,\"journal\":{\"name\":\"St Petersburg Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"St Petersburg Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/spmj/1769\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1769","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Hölder classes in the 𝐿^{𝑝} norm on a chord-arc curve in ℝ³
The Hölder classes Lpα(L)L_p^{\alpha } (L) in the Lp(L)L_p(L) norm on a chord-arc curve LL in R3\mathbb {R}^3 are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the Lp(L)L^p(L) norm, the direct theorem is proved for a certain subclass of Lpα(L)L^\alpha _p(L) and the inverse theorem covers the entire Hölder class.
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.
京公网安备 11010802042870号
求助内容：
应助结果提醒方式：
