Let 𝒜{mathcal{A}} be a complex unital Banach algebra and let R⊆𝒜{Rsubseteqmathcal{A}} be a non-empty set. This paper defines the property such that R is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity R with (CID) property, RD{R^{D}} is constructed as an extension of R to axiomatically study the accumulation of σR(
设𝒜 {mathcal{A}} 是一个复单元巴纳赫代数，设 R ⊆ 𝒜 {Rsubseteqmathcal{A}} 是一个非空集。本文定义了 R 闭合幂分解的性质（简言之，（CID）性质），以探索谱分解关系。进一步，对于具有（CID）性质的上半圆性 R，构造 R D {R^{D}} 作为 R 的扩展，以公理地研究任意 a∈ 𝒜 {ainmathcal{A}} 的 σ R ( a ) {sigma_{R}(a)} 的累积。 .最后，还提供了几个关于巴拿赫代数和算子代数的示例。
{"title":"Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum","authors":"Ying Liu, Li Jiang","doi":"10.1515/forum-2023-0376","DOIUrl":"https://doi.org/10.1515/forum-2023-0376","url":null,"abstract":"\u0000 <jats:p>Let <jats:inline-formula id=\"j_forum-2023-0376_ineq_9999\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi mathvariant=\"script\">𝒜</m:mi>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0376_eq_0364.png\" />\u0000 <jats:tex-math>{mathcal{A}}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> be a complex unital Banach algebra and let <jats:inline-formula id=\"j_forum-2023-0376_ineq_9998\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>R</m:mi>\u0000 <m:mo>⊆</m:mo>\u0000 <m:mi mathvariant=\"script\">𝒜</m:mi>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0376_eq_0275.png\" />\u0000 <jats:tex-math>{Rsubseteqmathcal{A}}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> be a non-empty set. This paper defines the property such that <jats:italic>R</jats:italic> is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity <jats:italic>R</jats:italic> with (CID) property, <jats:inline-formula id=\"j_forum-2023-0376_ineq_9997\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mi>R</m:mi>\u0000 <m:mi>D</m:mi>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0376_eq_0278.png\" />\u0000 <jats:tex-math>{R^{D}}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> is constructed as an extension of <jats:italic>R</jats:italic> to axiomatically study the accumulation of <jats:inline-formula id=\"j_forum-2023-0376_ineq_9996\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:msub>\u0000 <m:mi>σ</m:mi>\u0000 <m:mi>R</m:mi>\u0000 </m:msub>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 ","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140661001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 L^{2} -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 𝑛-dimensional generalized Zakharov–Kuznetsov equation with n ≥ 3 ngeq 3 .
本文致力于研究广义扎哈罗夫-库兹涅佐夫方程的点收敛问题和非线性平滑问题。首先，我们提出了 Linares 和 Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov-Kuznetsov equation, SIAM J. Math. Analog.Anal.53 (2021), 1, 914-936] 以及 Linares 和 Ramos 的定理 1.8 [The Cauchy problem for the L 2 L^{2} - critical generalized Zakharov-Kuznetsov equation, SIAM J. Math. Anal. Cauchy problem for the L 2 L^{2} -critical generalized Zakharov-Kuznetsov equation in dimension 3, Comm.其次，我们给出了 Ribaud 和 Vento [A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations, C. R. Math.Math.Acad.第三，我们给出了 Linares 和 Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov-Kuznetsov equation, SIAM J. Math. Anal.Anal.最后，我们研究了 n ≥ 3 ngeq 3 的 𝑛 维广义扎哈罗夫-库兹涅佐夫方程的非线性平滑和均匀收敛性。
{"title":"Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation","authors":"Wei Yan, Weimin Wang, Xiangqian Yan","doi":"10.1515/forum-2023-0463","DOIUrl":"https://doi.org/10.1515/forum-2023-0463","url":null,"abstract":"\u0000 This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation.\u0000Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal.\u0000 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the \u0000 \u0000 \u0000 \u0000 L\u0000 2\u0000 \u0000 \u0000 \u0000 L^{2}\u0000 \u0000 -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations\u0000 46 (2021), 9, 1601–1627].\u0000Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris\u0000 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation.\u0000Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal.\u0000 53 (2021), 1, 914–936].\u0000Finally, we study the nonlinear smoothing and uniform convergence of 𝑛-dimensional generalized Zakharov–Kuznetsov equation with \u0000 \u0000 \u0000 \u0000 n\u0000 ≥\u0000 3\u0000 \u0000 \u0000 \u0000 ngeq 3\u0000 \u0000 .","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140663579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of K(z,z){K(z,z)} (where K(z,w){K(z,w)} is an automorphic kernel function) over the fundamental domain, one considers the integral of K(z,z)
第一位作者于 1999 年证明了塞尔伯格迹线公式的某种广义化。在这一推广中，不再考虑 K ( z , z ) 的积分 {K(z,z)}的积分（其中 K ( z , w ) {K(z,w)}是一个自动核函数）上的积分，而是考虑 K ( z , z ) 的积分 u ( z ) {K(z,z)u(z)} ，其中 u ( z ) {u(z)} 是拉普拉斯算子的固定自动特征函数。这个公式是为 PSL ( 2 , ℝ ) 的离散子群证明的 {mathrm{PSL}(2,mathbb{R})} 正如经典的塞尔伯格迹线公式一样，它是通过两种不同的方法（"几何 "和 "光谱"）求 K ( z , z ) 的积分而得到的 u ( z ) {K(z,z)u(z)} 。
{"title":"On the geometric trace of a generalized Selberg trace formula","authors":"András Biró, Dávid Tóth","doi":"10.1515/forum-2023-0344","DOIUrl":"https://doi.org/10.1515/forum-2023-0344","url":null,"abstract":"\u0000 <jats:p>A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of <jats:inline-formula id=\"j_forum-2023-0344_ineq_9999\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>K</m:mi>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mi>z</m:mi>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>z</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0344_eq_0456.png\" />\u0000 <jats:tex-math>{K(z,z)}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> (where <jats:inline-formula id=\"j_forum-2023-0344_ineq_9998\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>K</m:mi>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mi>z</m:mi>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>w</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0344_eq_0454.png\" />\u0000 <jats:tex-math>{K(z,w)}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> is an automorphic kernel function) over the fundamental domain, one considers the integral of <jats:inline-formula id=\"j_forum-2023-0344_ineq_9997\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>K</m:mi>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mi>z</m:mi>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>z</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 <m:m","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140662296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the geodesic orbit Randers spaces of the form (G/H,F){(G/H,F)}, such that G is one of the compact classical Lie groups SO(n){{mathrm{S}}{mathrm{O}}(n)}, SU(n){{mathrm{S}}{mathrm{U}}(n)}
本文将研究 ( G / H , F ) 形式的大地轨道兰德斯空间 {(G/H,F)} 。 {(G/H,F)}，使得 G 是紧凑经典李群 SO ( n ) {{mathrm{S}}{mathrm{O}}(n)} , SU ( n ) {{mathrm{S}}{mathrm{U}}(n)} , Sp ( n ) {{mathrm{S}}{mathrm{p}}(n)} 中的一个，而 H 是对角嵌入积 H 1 × ⋯ × H s {H_{1}timescdotstimes H_{s}} 。 这类空间包括球形、斯蒂费尔流形、格拉斯曼流形和旗流形。本研究是对大地轨道兰德斯空间 ( G / H , F ) 研究的贡献 {(G/H,F)}，H 为半简单。我们在广义 Stiefel 流形上的同质束中构建了非黎曼 Randers g.o. 度量的新范例，这些范例不是自然还原的。此外，我们还得到了这些 Randers g.o. 度量的具体表达式。
{"title":"Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds","authors":"Shaoxiang Zhang, Huibin Chen","doi":"10.1515/forum-2023-0256","DOIUrl":"https://doi.org/10.1515/forum-2023-0256","url":null,"abstract":"\u0000 <jats:p>In this article, we study the geodesic orbit Randers spaces of the form <jats:inline-formula id=\"j_forum-2023-0256_ineq_9999\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>G</m:mi>\u0000 <m:mo>/</m:mo>\u0000 <m:mi>H</m:mi>\u0000 </m:mrow>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>F</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0256_eq_0211.png\" />\u0000 <jats:tex-math>{(G/H,F)}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, such that <jats:italic>G</jats:italic> is one of the compact classical Lie groups <jats:inline-formula id=\"j_forum-2023-0256_ineq_9998\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>SO</m:mi>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mi>n</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0256_eq_0452.png\" />\u0000 <jats:tex-math>{{mathrm{S}}{mathrm{O}}(n)}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, <jats:inline-formula id=\"j_forum-2023-0256_ineq_9997\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>SU</m:mi>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mi>n</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0256_eq_0453.png\" />\u0000 <jats:tex-math>{{mathrm{S}}{mathrm{U}}(n)}</jats:tex-math>\u0000 ","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140662057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let p be a prime number. One of the most difficult and important questions in Galois theory these days is determining which pro-p groups can occur as maximal pro-p Galois groups. Some restrictions of the structure of maximal pro-p Galois groups are already known, and therefore, the families of all pro-p groups satisfying these restrictions are in the center of the research. In the current paper we deal with three of these families: The family of all pro-p groups which satisfy hereditarily the n-vanishing Massey product property, the family of 1-cyclotomic oriented pro-p groups, and the family of pro-p groups that are hereditarily of p-absolute Galois type. We show that all these families are closed under taking profinite completion of each order.
假设 p 是一个素数。目前伽罗瓦理论中最困难也是最重要的问题之一，就是确定哪些原 p 群可以作为最大原 p 伽罗瓦群出现。人们已经知道最大原 p 伽罗瓦群结构的一些限制条件，因此，满足这些限制条件的所有原 p 群族就成了研究的中心。本文将讨论其中的三个族：满足继承性 n 变马西积性质的所有原 p 群族，1-环原子定向原 p 群族，以及继承性 p 绝对伽罗瓦类型的原 p 群族。我们证明了所有这些族在取每个阶的无限完成时都是封闭的。
{"title":"Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion","authors":"Tamar Bar‐On","doi":"10.1515/forum-2023-0227","DOIUrl":"https://doi.org/10.1515/forum-2023-0227","url":null,"abstract":"\u0000 Let p be a prime number. One of the most difficult and important questions in Galois theory these days is determining which pro-p groups can occur as maximal pro-p Galois groups. Some restrictions of the structure of maximal pro-p Galois groups are already known, and therefore, the families of all pro-p groups satisfying these restrictions are in the center of the research. In the current paper we deal with three of these families: The family of all pro-p groups which satisfy hereditarily the n-vanishing Massey product property, the family of 1-cyclotomic oriented pro-p groups, and the family of pro-p groups that are hereditarily of p-absolute Galois type. We show that all these families are closed under taking profinite completion of each order.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140664777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We classify nef vector bundles on a smooth hyperquadric of dimension ≥4{geq 4} with first Chern class two over an algebraically closed field of characteristic zero.
{"title":"Nef vector bundles on a hyperquadric with first Chern class two","authors":"Masahiro Ohno","doi":"10.1515/forum-2023-0459","DOIUrl":"https://doi.org/10.1515/forum-2023-0459","url":null,"abstract":"\u0000 <jats:p>We classify nef vector bundles on a smooth hyperquadric of dimension <jats:inline-formula id=\"j_forum-2023-0459_ineq_9999\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi />\u0000 <m:mo>≥</m:mo>\u0000 <m:mn>4</m:mn>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0459_eq_0365.png\" />\u0000 <jats:tex-math>{geq 4}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> with first Chern class two\u0000over an algebraically closed field of characteristic zero.</jats:p>","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140663580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This corrigendum corrects Proposition 5.2 in [A. Dasgupta, L. Mohan and S. S. Mondal, Multilinear Fourier Integral operators on modulation spaces, Forum Math. 2024, 10.1515/forum-2023-0158].
本更正纠正了 [A. Dasgupta, L. Mohan and S. S. Mondal, Multilinear Fourier Integral operators on modulation spaces, Forum Math.Dasgupta、L. Mohan 和 S. S. Mondal，调制空间上的多线性傅里叶积分算子，Forum Math.2024, 10.1515/forum-2023-0158].
{"title":"Multilinear fourier integral operators on modulation spaces","authors":"Aparajita Dasgupta, Lalit Mohan, Shyam Swarup Mondal","doi":"10.1515/forum-2024-0088","DOIUrl":"https://doi.org/10.1515/forum-2024-0088","url":null,"abstract":"This corrigendum corrects Proposition 5.2 in [A. Dasgupta, L. Mohan and S. S. Mondal, Multilinear Fourier Integral operators on modulation spaces, Forum Math. 2024, 10.1515/forum-2023-0158].","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved.
{"title":"Weighted bilinear multiplier theorems in Dunkl setting via singular integrals","authors":"Suman Mukherjee, Sanjay Parui","doi":"10.1515/forum-2023-0398","DOIUrl":"https://doi.org/10.1515/forum-2023-0398","url":null,"abstract":"The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini
Let G(g;x):=∑n≤xg(n){G(g;x):=sum_{nleq x}g(n)} be the summatory function of an arithmetical function g(n){g(n)}. In this paper, we prove that we can write weighted averages of an arbitrary fixed number N of arithmetical functions gj(n),j∈{1,…,N}{g_{j}(n),,jin{1,dots,N}} as an integral involving the convolution (in the sense of Laplace) of Gj(x){G_{j}(x)}, j∈
设 G ( g ; x ) := ∑ n ≤ x g ( n ) {G(g;x):=sum_{nleq x}g(n)} 为算术函数 g ( n ) {g(n)} 的求和函数。本文将证明，我们可以写出任意固定数量 N 的算术函数 g j ( n ) , j ∈ { 1 , ... , N } 的加权平均数 {g_{j}(n),,jin{1,dots,N}} 是一个涉及 G j ( x ) {G_{j}(x)} 的卷积（拉普拉斯意义上）的积分，j∈ { 1 , ... , N }。 {jin{1,dots,N}} . .此外，我们还证明了一个特性，它使我们能够以非常简单自然的方式获得关于算术函数平均数的已知结果，并克服了一些著名问题的技术限制。
{"title":"Laplace convolutions of weighted averages of arithmetical functions","authors":"Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini","doi":"10.1515/forum-2023-0259","DOIUrl":"https://doi.org/10.1515/forum-2023-0259","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>g</m:mi> <m:mo>;</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msub> <m:mo largeop=\"true\" symmetric=\"true\">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0246.png\" /> <jats:tex-math>{G(g;x):=sum_{nleq x}g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the summatory function of an arithmetical function <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0403.png\" /> <jats:tex-math>{g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we prove that we can write weighted averages of an arbitrary fixed number <jats:italic>N</jats:italic> of arithmetical functions <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>g</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\"4.2pt\">,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0414.png\" /> <jats:tex-math>{g_{j}(n),,jin{1,dots,N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> as an integral involving the convolution (in the sense of Laplace) of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0257.png\" /> <jats:tex-math>{G_{j}(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>j</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for each abelian number field K of sufficiently large degree d there exists an element α∈K{alphain K} with K=ℚ(α){K=mathbb{Q}(alpha)} and absolute Weil height H(α)≪d|ΔK|12d{H(alpha)ll_{d}|Delta_{K}|^{frac{1}{2d}}}, where ΔK{Delta_{K}} denotes the discriminant of K. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent 12d{frac{1}{2d}} is best-possible when d is even.
我们证明，对于每个阶数为 d 的无性数域 K，都存在一个元素 α∈K {alphain K} ，其中 K = ℚ ( α ) {K=mathbb{Q}(alpha)} 且绝对韦尔高 H ( α ) ≪ d | Δ K | 1 2 d {H(alpha)ll_{d}|Delta_{K}|^{frac{1}{2d}} ，其中 Δ K {Delta_{K}} 表示 K 的判别式。 其中 Δ K {Delta_{K}} 表示 K 的判别式。这回答了鲁珀特在 1998 年提出的一个问题，即在阶数足够大的无性扩展的情况下。我们还证明了当 d 为偶数时，指数 1 2 d {frac{1}{2d}} 是最可能的。
{"title":"Small generators of abelian number fields","authors":"Martin Widmer","doi":"10.1515/forum-2023-0467","DOIUrl":"https://doi.org/10.1515/forum-2023-0467","url":null,"abstract":"We show that for each abelian number field <jats:italic>K</jats:italic> of sufficiently large degree <jats:italic>d</jats:italic> there exists an element <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0135.png\" /> <jats:tex-math>{alphain K}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>K</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>ℚ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0100.png\" /> <jats:tex-math>{K=mathbb{Q}(alpha)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and absolute Weil height <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:msub> <m:mo>≪</m:mo> <m:mi>d</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>K</m:mi> </m:msub> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>d</m:mi> </m:mrow> </m:mfrac> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0091.png\" /> <jats:tex-math>{H(alpha)ll_{d}|Delta_{K}|^{frac{1}{2d}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>K</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0128.png\" /> <jats:tex-math>{Delta_{K}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the discriminant of <jats:italic>K</jats:italic>. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>d</m:mi> </m:mrow> </m:mfrac> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0152.png\" /> <jats:tex-math>{frac{1}{2d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is best-possible when <jats:italic>d</jats:italic> is even.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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