{"title":"多各向异性核的估计及其在嵌入定理中的应用","authors":"Garnik Karapetyan, Mikael Arakelian","doi":"10.1016/j.trmi.2016.12.005","DOIUrl":null,"url":null,"abstract":"<div><p>In the current paper we consider an integral representation of functions and embedding theorems of multianisotropic Sobolev spaces in the three-dimensional case when the completely regular polyhedron has an arbitrary number of anisotropic vertices. This work generalizes results obtained in Karapetyan (in press) and Karapetyan (2016). Particularly, in Karapetyan (in press) the two-dimensional case was fully solved and in Karapetyan (2016) analogous results were obtained for the case of one anisotropic vertex. The problem takes root from various works of Sobolev, particularly, Sobolev (1938) and Sobolev (0000) <span>[4]</span>, <span>[5]</span>. Related results were obtained by many authors and can be found in Besov et al. (1967), Reshetnyak (1971), Smith (1961), Nikolsky (0000) and Il’in (1967) <span>[6]</span>, <span>[7]</span>, <span>[8]</span>, <span>[9]</span>, <span>[10]</span>. The monograph (Besov, 1978) contains an overview of the problem. The results obtained in this paper are based on a generalization of regularization by a quasi-homogeneous polynomial (see Uspenskii (1972) and Karapetyan (1990) <span>[11]</span>, <span>[12]</span>).</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2017-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2016.12.005","citationCount":"1","resultStr":"{\"title\":\"Estimation of multianisotropic kernels and their application to the embedding theorems\",\"authors\":\"Garnik Karapetyan, Mikael Arakelian\",\"doi\":\"10.1016/j.trmi.2016.12.005\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In the current paper we consider an integral representation of functions and embedding theorems of multianisotropic Sobolev spaces in the three-dimensional case when the completely regular polyhedron has an arbitrary number of anisotropic vertices. This work generalizes results obtained in Karapetyan (in press) and Karapetyan (2016). Particularly, in Karapetyan (in press) the two-dimensional case was fully solved and in Karapetyan (2016) analogous results were obtained for the case of one anisotropic vertex. The problem takes root from various works of Sobolev, particularly, Sobolev (1938) and Sobolev (0000) <span>[4]</span>, <span>[5]</span>. Related results were obtained by many authors and can be found in Besov et al. (1967), Reshetnyak (1971), Smith (1961), Nikolsky (0000) and Il’in (1967) <span>[6]</span>, <span>[7]</span>, <span>[8]</span>, <span>[9]</span>, <span>[10]</span>. The monograph (Besov, 1978) contains an overview of the problem. The results obtained in this paper are based on a generalization of regularization by a quasi-homogeneous polynomial (see Uspenskii (1972) and Karapetyan (1990) <span>[11]</span>, <span>[12]</span>).</p></div>\",\"PeriodicalId\":43623,\"journal\":{\"name\":\"Transactions of A Razmadze Mathematical Institute\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2017-04-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.trmi.2016.12.005\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of A Razmadze Mathematical Institute\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2346809216301039\",\"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":"Transactions of A Razmadze Mathematical Institute","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2346809216301039","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: 1
摘要
在完全正多面体具有任意数目的各向异性顶点的情况下,本文研究了三维情况下多各向异性Sobolev空间的函数的积分表示和嵌入定理。这项工作概括了Karapetyan(出版中)和Karapetyan(2016)中获得的结果。特别是,在Karapetyan (in press)中,二维情况得到了完全解决,在Karapetyan(2016)中,对于一个各向异性顶点的情况得到了类似的结果。这个问题的根源在于Sobolev(1938)和Sobolev(0000)[4],[5]。许多作者都得到了相关的结果,如Besov et al.(1967)、Reshetnyak(1971)、Smith(1961)、Nikolsky(0000)和Il 'in(1967)[6]、[7]、[8]、[9]、[10]。专著(Besov, 1978)包含了对这个问题的概述。本文得到的结果是基于准齐次多项式对正则化的推广(见Uspenskii(1972)和Karapetyan(1990)[11],[12])。
Estimation of multianisotropic kernels and their application to the embedding theorems
In the current paper we consider an integral representation of functions and embedding theorems of multianisotropic Sobolev spaces in the three-dimensional case when the completely regular polyhedron has an arbitrary number of anisotropic vertices. This work generalizes results obtained in Karapetyan (in press) and Karapetyan (2016). Particularly, in Karapetyan (in press) the two-dimensional case was fully solved and in Karapetyan (2016) analogous results were obtained for the case of one anisotropic vertex. The problem takes root from various works of Sobolev, particularly, Sobolev (1938) and Sobolev (0000) [4], [5]. Related results were obtained by many authors and can be found in Besov et al. (1967), Reshetnyak (1971), Smith (1961), Nikolsky (0000) and Il’in (1967) [6], [7], [8], [9], [10]. The monograph (Besov, 1978) contains an overview of the problem. The results obtained in this paper are based on a generalization of regularization by a quasi-homogeneous polynomial (see Uspenskii (1972) and Karapetyan (1990) [11], [12]).