本文讨论了粗糙傅立叶积分算子 T a , φ T_{a,varphi} 的 L p L^{p} 有界性，其振幅 a ( x , ξ ) ∈ L ∞ S ρ m a(x、xi)in L^{infty}S_{rho}^{m} 和相位函数 φ ( x , ξ ) ∈ L ∞ Φ 2 varphi(x,xi)in{L^{infty}}{Phi^{2}} ，它满足一个度量条件。如果 m < n ( ρ - 1 ) p - ρ ( n - 1 ) 2 p m<；frac{n(rho-1)}{p}-frac{rho(n-1)}{2p} when 1 ≤ p ≤ 2 1leq pleq 2 or m < n ( ρ - 1 ) 2 - ρ ( n - 1 ) 2 ( 1 - 1 p ) m<；frac{n(rho-1)}{2}-frac{rho(n-1)}{2}(1-frac{1}{p}) when 2 ≤ p ≤ ∞ 2leq pleqinfty.我们的主要结果扩展并改进了关于傅里叶积分算子 L p L^{p} 有界性的一些已知结果。
{"title":"On 𝐿𝑝 boundedness of rough Fourier integral operators","authors":"Guoning Wu, Jie Yang","doi":"10.1515/forum-2023-0443","DOIUrl":"https://doi.org/10.1515/forum-2023-0443","url":null,"abstract":"In this paper, we deal with the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</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-2023-0443_ineq_0001.png\"/> <jats:tex-math>L^{p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> boundedness of rough Fourier integral operators <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>φ</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0443_ineq_0002.png\"/> <jats:tex-math>T_{a,varphi}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with amplitude <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>a</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant=\"normal\">∞</m:mi> </m:msup> <m:mo></m:mo> <m:msubsup> <m:mi>S</m:mi> <m:mi>ρ</m:mi> <m:mi>m</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0443_ineq_0003.png\"/> <jats:tex-math>a(x,xi)in L^{infty}S_{rho}^{m}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and phase function <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>φ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant=\"normal\">∞</m:mi> </m:msup> <m:mo></m:mo> <m:msup> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0443_ineq_0004.png\"/> <jats:tex-math>varphi(x,xi)in{L^{infty}}{Phi^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> which satisfies a measure condition. We show that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>φ</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0443_ineq_0002.png\"/> <jats:tex-math>T_{a,varphi}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is bounded on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/19","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883148","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 r⩾3rgeqslant 3 be an integer and 𝑄 any positive definite quadratic form in 𝑟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces SQS_{Q} defined by x3=Q(y1,…,yr)zx^{3}=Q(y_{1},dots,y_{r})z. This confirms Manin’s conjecture for any SQS_{Q}. Our proof is based on analytic methods, and uses some estimates for character sums and moments of 𝐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r=3r=3.
设 r ⩾ 3 rgeqslant 3 为整数，𝑄 为 𝑟 变量中的任意正定二次型。我们用省力误差项建立了奇异超曲面 S Q S_{Q} 上有界高的有理点数的渐近公式，定义为 x 3 = Q ( y 1 , ... , y r ) z x^{3}=Q(y_{1},dots,y_{r})z 。这证实了马宁对任意 S Q S_{Q} 的猜想。我们的证明基于分析方法，并使用了𝐿 函数的特征和与矩的一些估计值。特别是，其中一个要素是西格尔的质量公式，它是针对 r = 3 r=3 情况的论证。
{"title":"Rational points on a class of cubic hypersurfaces","authors":"Yujiao Jiang, Tingting Wen, Wenjia Zhao","doi":"10.1515/forum-2023-0394","DOIUrl":"https://doi.org/10.1515/forum-2023-0394","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>r</m:mi> <m:mo>⩾</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0394_ineq_0001.png\"/> <jats:tex-math>rgeqslant 3</jats:tex-math> </jats:alternatives> </jats:inline-formula> be an integer and 𝑄 any positive definite quadratic form in 𝑟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>S</m:mi> <m:mi>Q</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0394_ineq_0002.png\"/> <jats:tex-math>S_{Q}</jats:tex-math> </jats:alternatives> </jats:inline-formula> defined by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:mi>Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>y</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>y</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>z</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0394_ineq_0003.png\"/> <jats:tex-math>x^{3}=Q(y_{1},dots,y_{r})z</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This confirms Manin’s conjecture for any <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>S</m:mi> <m:mi>Q</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0394_ineq_0002.png\"/> <jats:tex-math>S_{Q}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Our proof is based on analytic methods, and uses some estimates for character sums and moments of 𝐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0394_ineq_0005.png\"/> <jats:tex-math>r=3</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785631","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 prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1<p≤q<∞1<pleq q<infty is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.
{"title":"On fractional inequalities on metric measure spaces with polar decomposition","authors":"Aidyn Kassymov, Michael Ruzhansky, Gulnur Zaur","doi":"10.1515/forum-2024-0056","DOIUrl":"https://doi.org/10.1515/forum-2024-0056","url":null,"abstract":"In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for <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>p</m:mi> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0056_ineq_0001.png\"/> <jats:tex-math>1<pleq q<infty</jats:tex-math> </jats:alternatives> </jats:inline-formula> is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609162","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 L be an even lattice of odd rank with discriminant group L′/L{L^{prime}/L}, and let α,β∈L′/L{alpha,betain L^{prime}/L}. We prove the Weil bound for the Kloosterman sums Sα,β(m,n,c){S_{alpha,beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.
让 L 是奇数阶的偶数网格，其判别群为 L ′ / L {L^{prime}/L} ，并让α , β ∈ L ′ / L {alpha,betain L^{prime}/L} . 让 α , β ∈ L ′ / L {L^{prime}/L} 。我们通过证明一个将 Kloosterman 和的除数和与稀疏指数和相关联的同一性来得到这个边界。这一特性概括了科南特性（Kohnen's identity for plus space Kloosterman sums with theta multiplier system）。
{"title":"The Weil bound for generalized Kloosterman sums of half-integral weight","authors":"Nickolas Andersen, Gradin Anderson, Amy Woodall","doi":"10.1515/forum-2023-0367","DOIUrl":"https://doi.org/10.1515/forum-2023-0367","url":null,"abstract":"Let <jats:italic>L</jats:italic> be an even lattice of odd rank with discriminant group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>/</m:mo> <m:mi>L</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0367_eq_0327.png\"/> <jats:tex-math>{L^{prime}/L}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>/</m:mo> <m:mi>L</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0367_eq_0384.png\"/> <jats:tex-math>{alpha,betain L^{prime}/L}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove the Weil bound for the Kloosterman sums <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>c</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-2023-0367_eq_0360.png\"/> <jats:tex-math>{S_{alpha,beta}(m,n,c)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of half-integral weight for the Weil Representation attached to <jats:italic>L</jats:italic>. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513636","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 prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori 𝕋d{mathbb{T}^{d}}, where d≥3{dgeq 3}. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter, The proof of the l2l^{2} decoupling conjecture, Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong, Strichartz estimates for N-body Schrödinger operators with small potential interactions, Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.
在本文中，我们证明了周期性背景下多体薛定谔方程的斯特里查茨估计，特别是在𝕋 d {mathbb{T}^{d}} 的环上。 其中 d ≥ 3 {dgeq 3} 。这些结果对有理和无理环都成立，而且在一定意义上对小的相互作用势也成立。我们的工作基于周期域上薛定谔算子的标准斯特里查兹估计 [J. Bourgain 和 C. Demeter]。Bourgain and C. Demeter, The proof of the l 2 l^{2} decoupling conjecture, Ann.作为比较，这一结果可被视为 [Y. Hong, Strichartz estimates for N.C.] 的周期性类似物。Hong, Strichartz estimates for N-body Schrödinger operators with small potential interactions, Discrete Contin.Dyn.Syst.37 2017, 10, 5355-5365]，尽管我们没有使用相同的扰动方法。我们还注意到，由于周期性 Strichartz 估计的导数损失特性，扰动方法失败了。
{"title":"On Strichartz estimates for many-body Schrödinger equation in the periodic setting","authors":"Xiaoqi Huang, Xueying Yu, Zehua Zhao, Jiqiang Zheng","doi":"10.1515/forum-2024-0105","DOIUrl":"https://doi.org/10.1515/forum-2024-0105","url":null,"abstract":"In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>𝕋</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0105_eq_0168.png\"/> <jats:tex-math>{mathbb{T}^{d}}</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:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0105_eq_0185.png\"/> <jats:tex-math>{dgeq 3}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter, The proof of the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>l</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0105_eq_0087.png\"/> <jats:tex-math>l^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> decoupling conjecture, Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong, Strichartz estimates for <jats:italic>N</jats:italic>-body Schrödinger operators with small potential interactions, Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"117 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513634","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 main aim of this paper is to study the arithmetic Bohr radius for holomorphic functions defined on a Reinhardt domain in ℂn{mathbb{C}^{n}} with positive real part. The present investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 2611–2619]. A part of our study in the present paper includes a connection between the classical Bohr radius and the arithmetic Bohr radius of unit ball in the Minkowski space ℓqn{ell^{n}_{q}}, 1≤q≤∞{1leq qleqinfty}. Further, we determine the exact value of a Bohr radius in terms of arithmetic Bohr radius.
本文的主要目的是研究定义在ℂ n {mathbb{C}^{n} 中具有正实部的莱因哈特域上的全形函数的算术玻尔半径。本研究受 Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 2611-2619] 的工作启发。本文研究的一部分包括经典玻尔半径与闵科夫斯基空间中单位球的算术玻尔半径 ℓ q n {ell^{n}_{q}} 之间的联系。 1 ≤ q ≤ ∞ {1leq qleqinfty} 。此外，我们用算术玻尔半径来确定玻尔半径的精确值。
{"title":"Arithmetic Bohr radius for the Minkowski space","authors":"Vasudevarao Allu, Himadri Halder, Subhadip Pal","doi":"10.1515/forum-2023-0425","DOIUrl":"https://doi.org/10.1515/forum-2023-0425","url":null,"abstract":"The main aim of this paper is to study the arithmetic Bohr radius for holomorphic functions defined on a Reinhardt domain in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℂ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0425_eq_0135.png\"/> <jats:tex-math>{mathbb{C}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with positive real part. The present investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 2611–2619]. A part of our study in the present paper includes a connection between the classical Bohr radius and the arithmetic Bohr radius of unit ball in the Minkowski space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi mathvariant=\"normal\">ℓ</m:mi> <m:mi>q</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0425_eq_0119.png\"/> <jats:tex-math>{ell^{n}_{q}}</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>q</m:mi> <m:mo>≤</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0425_eq_0078.png\"/> <jats:tex-math>{1leq qleqinfty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Further, we determine the exact value of a Bohr radius in terms of arithmetic Bohr radius.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513637","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 note, we define and study Asai gamma factors over finite fields. We also prove some results about local Asai L-functions over p-adic fields for level zero representations.
在本论文中，我们定义并研究了有限域上的浅井伽马因子。我们还证明了关于 p-adic 场上零级表示的局部 Asai L 函数的一些结果。
{"title":"Asai gamma factors over finite fields","authors":"Jingsong Chai","doi":"10.1515/forum-2024-0135","DOIUrl":"https://doi.org/10.1515/forum-2024-0135","url":null,"abstract":"In this note, we define and study Asai gamma factors over finite fields. We also prove some results about local Asai L-functions over p-adic fields for level zero representations.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513635","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 investigates the existence and multiplicity of solutions to fractional elliptic systems on conical spaces. Specifically, we focus on the challenges posed by complex geometric configurations, including cones with rough bases, and their implications for the treatment of lateral boundary conditions. By utilizing the fibering map approach and iterative method, we aim to address these challenges and provide new insights into the field. Notably, these issues have not been previously explored in existing literature, highlighting the originality and significance of our study.
{"title":"Multiple solutions for fractional elliptic systems","authors":"Zhao Guo","doi":"10.1515/forum-2023-0457","DOIUrl":"https://doi.org/10.1515/forum-2023-0457","url":null,"abstract":"This paper investigates the existence and multiplicity of solutions to fractional elliptic systems on conical spaces. Specifically, we focus on the challenges posed by complex geometric configurations, including cones with rough bases, and their implications for the treatment of lateral boundary conditions. By utilizing the fibering map approach and iterative method, we aim to address these challenges and provide new insights into the field. Notably, these issues have not been previously explored in existing literature, highlighting the originality and significance of our study.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"174 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513638","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}
Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.
相对罗塔-巴克斯特群是罗塔-巴克斯特群的广义化，最近在李群的背景下被引入。在本文中，我们探讨了相对罗塔-巴克斯特群与偏左括号的联系，众所周知，偏左括号给出了杨-巴克斯特方程的双射非退化集合理论解。我们证明了每一个相对 Rota-Baxter 群都会产生一个斜左撑，反之，每一个斜左撑都来自一个相对 Rota-Baxter 群。事实证明，在一些温和的限制条件下，这两个范畴之间存在同构关系。我们提出了一种高效的 GAP 算法，可以计算有限群上的相对 Rota-Baxter 算子。最后，我们引入了相对罗塔-巴克斯特群的等线性概念，并证明了这些对象的等线性诱导了相应斜左括号的等线性。
{"title":"Relative Rota–Baxter groups and skew left braces","authors":"Nishant Rathee, Mahender Singh","doi":"10.1515/forum-2024-0020","DOIUrl":"https://doi.org/10.1515/forum-2024-0020","url":null,"abstract":"Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"57 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503336","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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