{"title":"索波列夫空间中高阶导数的尖锐估计值","authors":"T. A. Garmanova, I. A. Sheipak","doi":"10.3103/s0027132224700013","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper describes the splines <span>\\(Q_{n,k}(x,a)\\)</span>, which define the relations <span>\\(y^{(k)}(a)=\\int\\limits_{0}^{1}y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx\\)</span> for an arbitrary point <span>\\(a\\in(0;1)\\)</span> and an arbitrary function <span>\\(y\\in\\mathring{W}^{n}_{p}[0;1]\\)</span>. The connection of the minimization of the norm <span>\\(\\|Q^{(n)}_{n,k}\\|_{L_{p^{\\prime}}[0;1]}\\)</span> (<span>\\(1/p+1/p^{\\prime}=1\\)</span>) by parameter <span>\\(a\\)</span> with the problem of best estimates for derivatives <span>\\(|y^{(k)}(a)|\\leqslant A_{n,k,p}(a)\\|y^{(n)}\\|_{L_{p}[0;1]}\\)</span>, and also with the problem of finding the exact embedding constants of the Sobolev space <span>\\(\\mathring{W}^{n}_{p}[0;1]\\)</span> into the space <span>\\(\\mathring{W}^{k}_{\\infty}[0;1]\\)</span>, <span>\\(n\\in\\mathbb{N}\\)</span>, <span>\\(0\\leqslant k\\leqslant n-1\\)</span>. Exact embedding constants are found for all <span>\\(n\\in\\mathbb{N}\\)</span>, <span>\\(k=n-1\\)</span> for <span>\\(p=1\\)</span> and for <span>\\(p=\\infty\\)</span>.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":null,"pages":null},"PeriodicalIF":0.2000,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sharp Estimates of High-Order Derivatives in Sobolev Spaces\",\"authors\":\"T. A. Garmanova, I. A. Sheipak\",\"doi\":\"10.3103/s0027132224700013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>The paper describes the splines <span>\\\\(Q_{n,k}(x,a)\\\\)</span>, which define the relations <span>\\\\(y^{(k)}(a)=\\\\int\\\\limits_{0}^{1}y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx\\\\)</span> for an arbitrary point <span>\\\\(a\\\\in(0;1)\\\\)</span> and an arbitrary function <span>\\\\(y\\\\in\\\\mathring{W}^{n}_{p}[0;1]\\\\)</span>. The connection of the minimization of the norm <span>\\\\(\\\\|Q^{(n)}_{n,k}\\\\|_{L_{p^{\\\\prime}}[0;1]}\\\\)</span> (<span>\\\\(1/p+1/p^{\\\\prime}=1\\\\)</span>) by parameter <span>\\\\(a\\\\)</span> with the problem of best estimates for derivatives <span>\\\\(|y^{(k)}(a)|\\\\leqslant A_{n,k,p}(a)\\\\|y^{(n)}\\\\|_{L_{p}[0;1]}\\\\)</span>, and also with the problem of finding the exact embedding constants of the Sobolev space <span>\\\\(\\\\mathring{W}^{n}_{p}[0;1]\\\\)</span> into the space <span>\\\\(\\\\mathring{W}^{k}_{\\\\infty}[0;1]\\\\)</span>, <span>\\\\(n\\\\in\\\\mathbb{N}\\\\)</span>, <span>\\\\(0\\\\leqslant k\\\\leqslant n-1\\\\)</span>. Exact embedding constants are found for all <span>\\\\(n\\\\in\\\\mathbb{N}\\\\)</span>, <span>\\\\(k=n-1\\\\)</span> for <span>\\\\(p=1\\\\)</span> and for <span>\\\\(p=\\\\infty\\\\)</span>.</p>\",\"PeriodicalId\":42963,\"journal\":{\"name\":\"Moscow University Mathematics Bulletin\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.2000,\"publicationDate\":\"2024-05-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Moscow University Mathematics Bulletin\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3103/s0027132224700013\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow University Mathematics Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s0027132224700013","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
Abstract The paper describes the splines \(Q_{n,k}(x,a)\), which define the relations\(y^{(k)}(a)=\int\limits_{0}^{1}y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx\) for an arbitrary point \(a\in(0. 1)\)和 an arbitrary function \(y\in\mathring{W}^{n}_{p}[0;1]\);1))和任意函数(y\in\mathring{W}^{n}_{p}[0;1]\).最小化规范 \(\|Q^{(n)}_{n,k}\|_{L_{p^{\prime}}[0;参数 \(a\) 的 (\(1/p+1/p^{\prime}=1\)) 与导数 \(|y^{(k)}(a)|leqslant A_{n,k,p}(a)\|y^{(n)}\|{L_{p}[0;1]})的问题,以及找到索波列夫空间 \(\mathring{W}^{n}_{p}[0;1]\) 到空间 \(\mathring{W}^{k}_{infty}[0;1]\), \(n\in\mathbb{N}\), \(0\leqslant k\leqslant n-1\) 的精确嵌入常数的问题。对于所有的(n\in\mathbb{N}\)、(k=n-1\)的(p=1\)和(p=\infty\),都可以找到精确的嵌入常数。
Sharp Estimates of High-Order Derivatives in Sobolev Spaces
Abstract
The paper describes the splines \(Q_{n,k}(x,a)\), which define the relations \(y^{(k)}(a)=\int\limits_{0}^{1}y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx\) for an arbitrary point \(a\in(0;1)\) and an arbitrary function \(y\in\mathring{W}^{n}_{p}[0;1]\). The connection of the minimization of the norm \(\|Q^{(n)}_{n,k}\|_{L_{p^{\prime}}[0;1]}\) (\(1/p+1/p^{\prime}=1\)) by parameter \(a\) with the problem of best estimates for derivatives \(|y^{(k)}(a)|\leqslant A_{n,k,p}(a)\|y^{(n)}\|_{L_{p}[0;1]}\), and also with the problem of finding the exact embedding constants of the Sobolev space \(\mathring{W}^{n}_{p}[0;1]\) into the space \(\mathring{W}^{k}_{\infty}[0;1]\), \(n\in\mathbb{N}\), \(0\leqslant k\leqslant n-1\). Exact embedding constants are found for all \(n\in\mathbb{N}\), \(k=n-1\) for \(p=1\) and for \(p=\infty\).
期刊介绍:
Moscow University Mathematics Bulletin is the journal of scientific publications reflecting the most important areas of mathematical studies at Lomonosov Moscow State University. The journal covers research in theory of functions, functional analysis, algebra, geometry, topology, ordinary and partial differential equations, probability theory, stochastic processes, mathematical statistics, optimal control, number theory, mathematical logic, theory of algorithms, discrete mathematics and computational mathematics.