通过 d n , v α = { v α σ ( α ) ( n ) 定义一个无穷矩阵 D α = ( d n , v α ) mathfrak{D}^{alpha}=(d^{alpha}_{n,v}) 、 v ∣ n , 0 , v ∤ n , d^{alpha}_{n,v}=begin{cases}dfrac{v^{alpha}}{sigma^{(alpha)}(n)},&；vmid n,（0,&；vnmid n,end{cases} 其中 σ ( α ) ( n ) sigma^{(alpha)}(n) 被定义为 n∈ N ninmathbb{N} 的正除数的𝛼次幂之和，并构造矩阵域 ℓ p ( D α ) ell_{p}(mathfrak{D}^{alpha}) ( 0 <；p < ∞ 0<p<；infty ), c 0 ( D α ) c_{0}(mathfrak{D}^{alpha}) , c ( D α ) c(mathfrak{D}^{alpha}) 和 ℓ ∞ ( D α ) ell_{infty}(mathfrak{D}^{alpha}) 由矩阵 D α mathfrak{D}^{alpha} 定义。我们建立了 Schauder 基，并确定了这些新空间的 𝛼-、 𝛼-和 𝛾-对偶。我们描述了从ℓ p ( D α ) ell_{p}(mathfrak{D}^{alpha}) , c 0 ( D α ) c_{0}(mathfrak{D}^{alpha}) 的矩阵变换、 c ( D α ) c(mathfrak{D}^{alpha}) and ℓ ∞ ( D α ) ell_{infty}(mathfrak{D}^{alpha}) to ℓ ∞ ell_{infty} , 𝑐, c 0 c_{0} 和 ℓ 1 ell_{1} 。此外，我们还确定了 X∈ { ℓ p ( D α ) , c 0 ( D α ) , c ( D α ) , ℓ ∞ ( D α ) } 的算子（或矩阵）紧凑性的一些标准。 Xin{ell_{p}(mathfrak{D}^{alpha}),c_{0}(mathfrak{D}^{alpha}),c(mathfrak{D}^{alpha}),ell_{infty}(mathfrak{D}^{alpha})} to ℓ ∞ ell_{infty} , 𝑐, c 0 c_{0} 或 ℓ 1 ell_{1} .
The moduli space of planar polygons with generic side lengths is a smooth, closed manifold. It is known that these manifolds contain the moduli space of distinct points on the real projective line as an open dense subset. Kapranov showed that the real points of the Deligne–Mumford–Knudson compactification can be obtained from the projective Coxeter complex of type 𝐴 (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper, we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type 𝐴 by performing an iterative cellular surgery along a subcollection of the minimal building set. Interestingly, this subcollection is determined by the combinatorial data associated with the length vector called the genetic code.
{"title":"Building planar polygon spaces from the projective braid arrangement","authors":"Navnath Daundkar, Priyavrat Deshpande","doi":"10.1515/forum-2023-0032","DOIUrl":"https://doi.org/10.1515/forum-2023-0032","url":null,"abstract":"The moduli space of planar polygons with generic side lengths is a smooth, closed manifold. It is known that these manifolds contain the moduli space of distinct points on the real projective line as an open dense subset. Kapranov showed that the real points of the Deligne–Mumford–Knudson compactification can be obtained from the projective Coxeter complex of type 𝐴 (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper, we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type 𝐴 by performing an iterative cellular surgery along a subcollection of the minimal building set. Interestingly, this subcollection is determined by the combinatorial data associated with the length vector called the genetic code.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"59 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002472","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}
在本文中，我们将研究鞅 cok ( A n + p x i I n ) {operatorname{cok}(A_{n}+px_{i}I_{n})} ( 1 ≤ i ≤ m {1leq ileq m} ) 之间的组合关系，其中 A n {A_{n}} 是 p-adic 整数环上的 n × n {ntimes n} 矩阵。 I n {I_{n}} 是 n × n {ntimes n} 的标识矩阵，x 1 , ... , x m {x_{1},dots,x_{m}} 是ℤ p {mathbb{Z}_{p}} 的元素，它们的还原模数 p 是不同的。对于正整数 m ≤ 4 {mleq 4} 并且给定 x 1 , ... , x m ∈ ℤ p {x_{1},dots,x_{m}inmathbb{Z}_{p}}, 我们可以确定 m-t 集。 ，我们确定有限生成的ℤ p {x_{1},dots,x_{m}inmathbb{Z}_{p}} 的 m 元组集合。 -模块 ( H 1 , ... , H m ) {(H_{1},dots,H_{m})} ，其中 ( cok ( A n + p x 1 I n ) , ... , cok ( A n + p x m I n ) ) = ( H 1 , ... , H m ) (operator) {(H_{1},dots,H_{m})} 。, H m ) (operatorname{cok}(A_{n}+px_{1}I_{n}),dots,operatorname{cok}(A_{n}+px_{m}I_% {n}))=(H_{1},dots,H_{m}) for some matrix A n {A_{n}}. .我们还可以证明，如果 A n {A_{n}} 是一个 n × n {ntimes n} 的哈尔随机矩阵。 则 cok ( A n + p x i I n ) {operatorname{cok}(A_{n}+px_{i}I_{n})} ( 1 ≤ i ≤ m {1leq ileq m} ) 的联合分布在 n → ∞ {nrightarrowinfty} 时收敛。
{"title":"Joint distribution of the cokernels of random p-adic matrices II","authors":"Jiwan Jung, Jungin Lee","doi":"10.1515/forum-2023-0131","DOIUrl":"https://doi.org/10.1515/forum-2023-0131","url":null,"abstract":"In this paper, we study the combinatorial relations between the cokernels <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>cok</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo></m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo></m:mo> <m:msub> <m:mi>I</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:mrow> <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-2023-0131_eq_0646.png\" /> <jats:tex-math>{operatorname{cok}(A_{n}+px_{i}I_{n})}</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:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>i</m:mi> <m:mo>≤</m:mo> <m:mi>m</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0131_eq_0323.png\" /> <jats:tex-math>{1leq ileq m}</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>A</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0131_eq_0378.png\" /> <jats:tex-math>{A_{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is an <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0131_eq_0825.png\" /> <jats:tex-math>{ntimes n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> matrix over the ring of <jats:italic>p</jats:italic>-adic integers <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0131_eq_0584.png\" /> <jats:tex-math>{mathbb{Z}_{p}}</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:msub> <m:mi>I</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0131_eq_0431.png\" /> <jats:tex-math>{I_{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"30 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920897","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 prove that the extended frame spectral measures are of pure type and the Beurling dimension of any frame measure for an extended frame spectral measure is in its Fourier dimension and upper entropy dimension.
{"title":"Some properties of extended frame measure","authors":"Jinjun Li, Zhiyi Wu, Fusheng Xiao","doi":"10.1515/forum-2023-0412","DOIUrl":"https://doi.org/10.1515/forum-2023-0412","url":null,"abstract":"We prove that the extended frame spectral measures are of pure type and the Beurling dimension of any frame measure for an extended frame spectral measure is in its Fourier dimension and upper entropy dimension.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"76 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920890","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}
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and Stäckel matrices.
{"title":"Orthogonal separation of variables for spaces of constant curvature","authors":"Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev","doi":"10.1515/forum-2023-0300","DOIUrl":"https://doi.org/10.1515/forum-2023-0300","url":null,"abstract":"We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and Stäckel matrices.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"51 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920893","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}
Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to x with implied constants depending on m and n. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in x, m and n. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of x with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in x, m and n. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to m and n, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function p(n){p(n)}. We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo 3, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.
克罗斯特曼和与模形式理论有很深的联系，对它们的估计有许多重要的结果。库兹涅佐夫（Kuznetsov）使用他著名的迹公式，得到了一个关于 x 的省力估计，其中隐含的常数取决于 m 和 n。Goldfeld 和 Sarnak 用对应于双曲拉普拉斯的特殊特征值的主项对它们的和进行了约束。他们的误差项是 x 的幂，其隐含常数取决于所有其他因子。在本文中，对于多种半整数权乘法器系统，我们得到了误差项均匀为 x、m 和 n 的相同约束。对于 eta 乘法器，Ahlgren 和 Andersen 得到了与 m 和 n 有关的均匀且省电的约束，从而得到了分区函数 p ( n ) {p(n)} 的拉德马赫精确公式的收敛误差估计值。我们还为秩模为 3 的分区差建立了一个拉德马赫式精确公式，从而可以将我们的省电估计应用于该公式的尾部，以获得收敛误差约束。
{"title":"Uniform bounds for Kloosterman sums of half-integral weight with applications","authors":"Qihang Sun","doi":"10.1515/forum-2023-0201","DOIUrl":"https://doi.org/10.1515/forum-2023-0201","url":null,"abstract":"Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to <jats:italic>x</jats:italic> with implied constants depending on <jats:italic>m</jats:italic> and <jats:italic>n</jats:italic>. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in <jats:italic>x</jats:italic>, <jats:italic>m</jats:italic> and <jats:italic>n</jats:italic>. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of <jats:italic>x</jats:italic> with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in <jats:italic>x</jats:italic>, <jats:italic>m</jats:italic> and <jats:italic>n</jats:italic>. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to <jats:italic>m</jats:italic> and <jats:italic>n</jats:italic>, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</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-2023-0201_eq_0934.png\" /> <jats:tex-math>{p(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo 3, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"238 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920962","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}
Employing Watson’s ϕ78{{}_{8}phi_{7}} transformation formula, we unearth several q-supercongruences with a parameter s. Particularly, one of our results is an extension of a q-analogue of Van Hamme’s (G.2) supercongruence. In addition, we obtain a q-supercongruence modulo the fifth power of a cyclotomic polynomial and propose two related conjectures.
{"title":"q-supercongruences from Watson's 8φ7 transformation","authors":"Xiaoxia Wang, Chang Xu","doi":"10.1515/forum-2023-0409","DOIUrl":"https://doi.org/10.1515/forum-2023-0409","url":null,"abstract":"Employing Watson’s <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mmultiscripts> <m:mi>ϕ</m:mi> <m:mn>7</m:mn> <m:none /> <m:mprescripts /> <m:mn>8</m:mn> <m:none /> </m:mmultiscripts> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0409_eq_0156.png\" /> <jats:tex-math>{{}_{8}phi_{7}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> transformation formula, we unearth several <jats:italic>q</jats:italic>-supercongruences with a parameter <jats:italic>s</jats:italic>. Particularly, one of our results is an extension of a <jats:italic>q</jats:italic>-analogue of Van Hamme’s (G.2) supercongruence. In addition, we obtain a <jats:italic>q</jats:italic>-supercongruence modulo the fifth power of a cyclotomic polynomial and propose two related conjectures.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920898","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 paper, we investigate certain submodules in C*-algebras associated to effective étale groupoids. First, we show that a submodule generated by normalizers is a closure of the set of compactly supported continuous functions on some open set. As a corollary, we show that discrete group coactions on groupoid C*-algebras are induced by cocycles of étale groupoids if the fixed point algebras contain C*-subalgebras of continuous functions vanishing at infinity on the unit spaces. In the latter part, we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras.
{"title":"Submodules of normalisers in groupoid C*-algebras and discrete group coactions","authors":"Fuyuta Komura","doi":"10.1515/forum-2023-0182","DOIUrl":"https://doi.org/10.1515/forum-2023-0182","url":null,"abstract":"In this paper, we investigate certain submodules in C*-algebras associated to effective étale groupoids. First, we show that a submodule generated by normalizers is a closure of the set of compactly supported continuous functions on some open set. As a corollary, we show that discrete group coactions on groupoid C*-algebras are induced by cocycles of étale groupoids if the fixed point algebras contain C*-subalgebras of continuous functions vanishing at infinity on the unit spaces. In the latter part, we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"190 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920859","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}
Der-Chen Chang, Xuan Thinh Duong, Ji Li, Wei Wang, Qingyan Wu
We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space Hp{H^{p}} on quaternionic Siegel upper half space for 23<p≤1{frac{2}{3}<pleq 1}. Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.
我们研究了四元西格尔上半空间的 Cauchy-Szegő 投影，得到了 Cauchy-Szegő 内核的点（高阶）正则性估计，并证明了 Cauchy-Szegő 内核处处非零，从而进一步得到了非退化的点下界。作为应用，我们证明了四元海森堡群上每个原子上的 Cauchy-Szegő 投影的均匀有界性，并用它给出了 2 3 < p ≤ 1 {frac{2}{3}<pleq 1} 时四元西格尔上半空间上正则哈代空间 H p {H^{p}} 的原子分解。此外，我们还基于核估计建立了考奇-塞格ő 投影换元的奇异值特征。四元数结构（缺乏换元性）被编码在正则函数的对称组和相关偏微分方程中。
{"title":"Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications","authors":"Der-Chen Chang, Xuan Thinh Duong, Ji Li, Wei Wang, Qingyan Wu","doi":"10.1515/forum-2024-0049","DOIUrl":"https://doi.org/10.1515/forum-2024-0049","url":null,"abstract":"We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>H</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0049_eq_0488.png\" /> <jats:tex-math>{H^{p}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on quaternionic Siegel upper half space for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mfrac> <m:mn>2</m:mn> <m:mn>3</m:mn> </m:mfrac> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0049_eq_0631.png\" /> <jats:tex-math>{frac{2}{3}<pleq 1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"34 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920899","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}
László Fuchs, Brendan Goldsmith, Luigi Salce, Lutz Strüngmann
Cellular covers which originate in homotopy theory are considered here for a very special class: divisible uniserial modules over valuation domains. This is a continuation of the study of cellular covers of divisible objects, but in order to obtain more substantial results, we are restricting our attention further to specific covers or to specific kernels. In particular, for h-divisible uniserial modules, we deal first with covers limited to divisible torsion-free modules (Section 3), and continue with the restriction to torsion standard uniserials (Sections 4–5). For divisible non-standard uniserial modules, only those cellular covers are investigated whose kernels are also divisible non-standard uniserials (Section 6). The results are specific enough to enable us to describe more accurately how to find all cellular covers obeying the chosen restrictions.
源于同调理论的蜂窝盖在这里针对一个非常特殊的类别进行了研究：可分的估值域上的单列模块。这是可分对象蜂窝盖研究的继续，但为了获得更多实质性结果，我们将注意力进一步限制在特定的盖或特定的核上。特别是，对于 h 可分的单列模块，我们首先处理限于可分的无扭模块的覆盖（第 3 节），然后继续处理限制于有扭标准单列模块的覆盖（第 4-5 节）。对于可分的非标准单偶数模块，我们只研究那些内核也是可分的非标准单偶数的单元盖（第 6 节）。这些结果足够具体，使我们能够更准确地描述如何找到所有符合所选限制条件的蜂窝盖。
{"title":"Cellular covers of divisible uniserial modules over valuation domains","authors":"László Fuchs, Brendan Goldsmith, Luigi Salce, Lutz Strüngmann","doi":"10.1515/forum-2023-0351","DOIUrl":"https://doi.org/10.1515/forum-2023-0351","url":null,"abstract":"Cellular covers which originate in homotopy theory are considered here for a very special class: divisible uniserial modules over valuation domains. This is a continuation of the study of cellular covers of divisible objects, but in order to obtain more substantial results, we are restricting our attention further to specific covers or to specific kernels. In particular, for <jats:italic>h</jats:italic>-divisible uniserial modules, we deal first with covers limited to divisible torsion-free modules (Section 3), and continue with the restriction to torsion standard uniserials (Sections 4–5). For divisible non-standard uniserial modules, only those cellular covers are investigated whose kernels are also divisible non-standard uniserials (Section 6). The results are specific enough to enable us to describe more accurately how to find all cellular covers obeying the chosen restrictions.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656233","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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