加权周期和离散伪微分算子

Aparajita Dasgupta, Lalit Mohan, Shyam Swarup Mondal
{"title":"加权周期和离散伪微分算子","authors":"Aparajita Dasgupta, Lalit Mohan, Shyam Swarup Mondal","doi":"10.1007/s00605-024-01976-w","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class <span>\\(M_{\\rho , \\Lambda }^m({\\mathbb {T}}\\times {\\mathbb {Z}})\\)</span> (associated to a suitable weight function <span>\\(\\Lambda \\)</span> on <span>\\({\\mathbb {Z}}\\)</span>) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of <i>M</i>-elliptic pseudo-differential operators on <span>\\({\\mathbb {T}}\\)</span>. Further, we prove a version of Gohberg’s lemma for pseudo-differetial operators with weighted symbol class <span>\\(M_{\\rho , \\Lambda }^0({\\mathbb {T}}\\times {\\mathbb {Z}})\\)</span> and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on <span>\\(L^2({\\mathbb {T}})\\)</span>. Finally, we provide Gårding’s and Sharp Gårding’s inequality for <i>M</i>-elliptic operators on <span>\\({\\mathbb {Z}}\\)</span> and <span>\\({\\mathbb {T}}\\)</span>, respectively, and present an application in the context of strong solution of the pseudo-differential equation <span>\\(T_{\\sigma } u=f\\)</span> in <span>\\(L^{2}\\left( {\\mathbb {T}}\\right) \\)</span>.\n</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Weighted periodic and discrete pseudo-differential Operators\",\"authors\":\"Aparajita Dasgupta, Lalit Mohan, Shyam Swarup Mondal\",\"doi\":\"10.1007/s00605-024-01976-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class <span>\\\\(M_{\\\\rho , \\\\Lambda }^m({\\\\mathbb {T}}\\\\times {\\\\mathbb {Z}})\\\\)</span> (associated to a suitable weight function <span>\\\\(\\\\Lambda \\\\)</span> on <span>\\\\({\\\\mathbb {Z}}\\\\)</span>) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of <i>M</i>-elliptic pseudo-differential operators on <span>\\\\({\\\\mathbb {T}}\\\\)</span>. Further, we prove a version of Gohberg’s lemma for pseudo-differetial operators with weighted symbol class <span>\\\\(M_{\\\\rho , \\\\Lambda }^0({\\\\mathbb {T}}\\\\times {\\\\mathbb {Z}})\\\\)</span> and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on <span>\\\\(L^2({\\\\mathbb {T}})\\\\)</span>. Finally, we provide Gårding’s and Sharp Gårding’s inequality for <i>M</i>-elliptic operators on <span>\\\\({\\\\mathbb {Z}}\\\\)</span> and <span>\\\\({\\\\mathbb {T}}\\\\)</span>, respectively, and present an application in the context of strong solution of the pseudo-differential equation <span>\\\\(T_{\\\\sigma } u=f\\\\)</span> in <span>\\\\(L^{2}\\\\left( {\\\\mathbb {T}}\\\\right) \\\\)</span>.\\n</p>\",\"PeriodicalId\":18913,\"journal\":{\"name\":\"Monatshefte für Mathematik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Monatshefte für Mathematik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00605-024-01976-w\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01976-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

在本文中,我们通过推导渐近和、组合、邻接、转置的公式,研究与加权符号类 \(M_{\rho , \Lambda }^m({\mathbb {T}}\times {\mathbb {Z}})\)(与\({\mathbb {Z}}\)上合适的权函数 \(\Lambda \)相关联)相关的伪微分算子的符号微积分要素。我们还构建了 M-elliptic 伪微分算子在 \({\mathbb {T}}\) 上的参数矩阵。此外,我们证明了加权符号类 \(M_{\rho , \Lambda }^0({\mathbb {T}}\times {\mathbb {Z}})\的伪微分算子的高伯格(Gohberg)定理的一个版本,并且作为应用,我们提供了一个充分必要条件来确保相应的伪微分算子在 \(L^2({\mathbb {T}})\) 上是紧凑的。最后,我们分别提供了M-椭圆算子在\({\mathbb {Z}}\) 和\({\mathbb {T}}\)上的高定不等式(Gårding's)和夏普高定不等式(Sharp Gårding's),并介绍了在\(L^{2}\left( {\mathbb {T}}\right) \)中伪微分方程\(T_{\sigma } u=f\)的强解中的应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Weighted periodic and discrete pseudo-differential Operators

In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class \(M_{\rho , \Lambda }^m({\mathbb {T}}\times {\mathbb {Z}})\) (associated to a suitable weight function \(\Lambda \) on \({\mathbb {Z}}\)) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of M-elliptic pseudo-differential operators on \({\mathbb {T}}\). Further, we prove a version of Gohberg’s lemma for pseudo-differetial operators with weighted symbol class \(M_{\rho , \Lambda }^0({\mathbb {T}}\times {\mathbb {Z}})\) and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on \(L^2({\mathbb {T}})\). Finally, we provide Gårding’s and Sharp Gårding’s inequality for M-elliptic operators on \({\mathbb {Z}}\) and \({\mathbb {T}}\), respectively, and present an application in the context of strong solution of the pseudo-differential equation \(T_{\sigma } u=f\) in \(L^{2}\left( {\mathbb {T}}\right) \).

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
期刊最新文献
On combinatorial properties of Gruenberg–Kegel graphs of finite groups Sparse bounds for oscillating multipliers on stratified groups Some sharp inequalities for norms in $$\mathbb {R}^n$$ and $$\mathbb {C}^n$$ Ill-posedness for the gCH-mCH equation in Besov spaces Stability of pseudo peakons for a new fifth order CH type equation with cubic nonlinearities
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1