Marjorie Drake, Charles Fefferman, Kevin Ren, Anna Skorobogatova
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</m:math> <jats:tex-math> $T :{L}^{2,p}\\left(E\\right)\\to {L}^{2,p}\\left({\\mathbb{R}}^{2}\\right)$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0132_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula> when 1 < <jats:italic>p</jats:italic> < 2 and <jats:italic>E</jats:italic> is a finite subset of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${\\mathbb{R}}^{2}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0132_ineq_002.png\"/> </jats:alternatives> </jats:inline-formula> contained in a line.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"83 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sobolev extension in a simple case\",\"authors\":\"Marjorie Drake, Charles Fefferman, Kevin Ren, Anna Skorobogatova\",\"doi\":\"10.1515/ans-2023-0132\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we establish the existence of a bounded, linear extension operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>T</m:mi> <m:mspace width=\\\"0.17em\\\"/> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:math> <jats:tex-math> $T :{L}^{2,p}\\\\left(E\\\\right)\\\\to {L}^{2,p}\\\\left({\\\\mathbb{R}}^{2}\\\\right)$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_ans-2023-0132_ineq_001.png\\\"/> </jats:alternatives> </jats:inline-formula> when 1 < <jats:italic>p</jats:italic> < 2 and <jats:italic>E</jats:italic> is a finite subset of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${\\\\mathbb{R}}^{2}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_ans-2023-0132_ineq_002.png\\\"/> </jats:alternatives> </jats:inline-formula> contained in a line.\",\"PeriodicalId\":7191,\"journal\":{\"name\":\"Advanced Nonlinear Studies\",\"volume\":\"83 1\",\"pages\":\"\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-05-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advanced Nonlinear Studies\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/ans-2023-0132\",\"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":"Advanced Nonlinear 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引用次数: 0
摘要
在本文中，当 1 < p < 2 且 E 是 R 2 的有限子集 ${\mathbb{R}}^{2}$ 包含在一条直线中时，我们建立了有界线性扩展算子 T : L 2 , p ( E ) → L 2 , p ( R 2 ) $T :{L}^{2,p}\left(E\right)/to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ 的存在性。
In this paper, we establish the existence of a bounded, linear extension operator T:L2,p(E)→L2,p(R2) $T :{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ when 1 < p < 2 and E is a finite subset of R2 ${\mathbb{R}}^{2}$ contained in a line.
期刊介绍：
Advanced Nonlinear Studies is aimed at publishing papers on nonlinear problems, particulalry those involving Differential Equations, Dynamical Systems, and related areas. It will also publish novel and interesting applications of these areas to problems in engineering and the sciences. Papers submitted to this journal must contain original, timely, and significant results. Articles will generally, but not always, be published in the order when the final copies were received.
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