Anilatmaja Aryasomayajula, Baskar Balasubramanyam, Dyuti Roy
In this article, for n≥2{ngeq 2}, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU((n,1),ℂ){mathrm{SU}((n,1),mathbb{C})}. The main result of the article is the following result. Let Γ⊂SU((2,1),𝒪K){Gammasubsetmathrm{SU}((2,1),mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬΓk{{{mathcal{B}_{Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮k(Γ)
在本文中，对于 n ≥ 2 {ngeq 2} ，我们计算了与 SU ( ( n , 1 ) , ℂ) 的无扭转、共偶子群相关的皮卡尔模块尖顶形式的伯格曼核的渐近、定性和定量估计值。 {mathrm{SU}((n,1),mathbb{C})}。文章的主要结果如下。设 Γ ⊂ SU ( ( 2 , 1 ) , 𝒪 K ) {Gammasubsetmathrm{SU}((2,1),mathcal{O}_{K})} 是一个有限索引的无扭子群，其中 K 是一个完全虚域。让 ℬ Γ k {{mathcal{B}_{Gamma}^{k}}}} 表示与 𝒮 k ( Γ ) {mathcal{S}_{k}(Gamma)} 相关的伯格曼核，它是关于 Γ 的权重-k 尖顶形式的复向量空间。让 𝔹 2 {mathbb{B}^{2}} 表示赋有双曲度量的二维复球，让 X Γ := Γ 𝔹 2 {X_{Gamma}:=Gammabackslashmathbb{B}^{2}} 表示商空间，它是维数为 2 的非紧凑复流形。让 |⋅ | pet {|cdot|_{mathrm{pet}}} 表示𝒮 k ( Γ ) 上的点向彼得森规范 {mathcal{S}_{k}(Gamma)} 。 我们有如下估计： sup z ∈ X Γ | ℬ Γ k ( z ) | pet = O Γ ( k 5 2 ) 、 |{{sup_{zin X_{Gamma}}|{{mathcal{B}_{Gamma}^{k}}}(z)|_{{mathrm{pet}}=O_{% Gamma}(k^{frac{5}{2}})，其中隐含的常数只取决于 Γ。
{"title":"Estimates of Picard modular cusp forms","authors":"Anilatmaja Aryasomayajula, Baskar Balasubramanyam, Dyuti Roy","doi":"10.1515/forum-2023-0079","DOIUrl":"https://doi.org/10.1515/forum-2023-0079","url":null,"abstract":"In this article, for <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:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0079_eq_0368.png\" /> <jats:tex-math>{ngeq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>SU</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <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-2023-0079_eq_0306.png\" /> <jats:tex-math>{mathrm{SU}((n,1),mathbb{C})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The main result of the article is the following result. Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo>⊂</m:mo> <m:mrow> <m:mi>SU</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mi mathvariant=\"script\">𝒪</m:mi> <m:mi>K</m:mi> </m:msub> <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-2023-0079_eq_0229.png\" /> <jats:tex-math>{Gammasubsetmathrm{SU}((2,1),mathcal{O}_{K})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a torsion-free subgroup of finite index, where <jats:italic>K</jats:italic> is a totally imaginary field. Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi mathvariant=\"script\">ℬ</m:mi> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mi>k</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0079_eq_0408.png\" /> <jats:tex-math>{{{mathcal{B}_{Gamma}^{k}}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the Bergman kernel associated to the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi mathvariant=\"script\">𝒮</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"normal\">Γ</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:h","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299677","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 first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in 𝔽q2{mathbb{F}_{q}^{2}}. More precisely, for A,B,C⊂𝔽q2{A,B,Csubsetmathbb{F}_{q}^{2}}, if |A||B||C|12≫q4{|A||B||C|^{frac{1}{2}}gg q^{4}}, then for any λ∈𝔽q∖{0}{lambdainmathbb{F}_{q}setminus{0}}, the number of congruence classes of triangles with vertices in A×B×C{Atimes Btimes C} and one side-l
本文的第一个目标是证明一个尖锐的条件，以保证在𝔽 q 2 {mathbb{F}_{q}^{2}} 中给定集合中所有全等类三角形的比例为正。 .更确切地说，对于 A , B , C ⊂ 𝔽 q 2 {A,B,Csubsetmathbb{F}_{q}^{2}} . 如果 | A | | B | | C | 1 2 ≫ q 4 {|A||B||C|^{frac{1}{2}}gg q^{4}} ，则 则对于任意 λ∈ 𝔽 q ∖ { 0 } {lambdainmathbb{F}_{q}setminus{0}} ，顶点在 A × B × C {Atimes Btimes C} 中且边长为 λ 的三角形的全等类的数目至少为 ≫ q 2 {gg q^{2}} 。 .在更高维度中，我们得到了 k-simplex 的类似结果，但条件稍强。与文献中著名的 L 2 {L^{2}} 方法相比，我们的方法在条件和结论上都提供了更好的结果。当 A = B = C {A=B=C} 时 本文的第二个目标是对 Bennett、Hart、Iosevich、Pakianathan 和 Rudnev (2017) 以及 McDonald (2020) 提出的关于单纯形分布的当前最佳结果给出新的统一证明。本文的第三个目标是研究与一组刚性运动相关的 Furstenberg 型问题。我们证明的主要内容是点与刚性运动之间的入射界限。大集合的入射边界由作者和 Semin Yoo (2023) 提出，而小集合的边界将通过使用 Kollár (2015) 提出的𝔽 q 3 {mathbb{F}_{q}^{3}} 中的点线入射边界来证明。
{"title":"Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces","authors":"Thang Pham","doi":"10.1515/forum-2023-0470","DOIUrl":"https://doi.org/10.1515/forum-2023-0470","url":null,"abstract":"The first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0470_eq_0275.png\" /> <jats:tex-math>{mathbb{F}_{q}^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. More precisely, for <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:mi>B</m:mi> <m:mo>,</m:mo> <m:mi>C</m:mi> </m:mrow> <m:mo>⊂</m:mo> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0470_eq_0164.png\" /> <jats:tex-math>{A,B,Csubsetmathbb{F}_{q}^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, if <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>C</m:mi> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> </m:mrow> <m:mo>≫</m:mo> <m:msup> <m:mi>q</m:mi> <m:mn>4</m:mn> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0470_eq_0450.png\" /> <jats:tex-math>{|A||B||C|^{frac{1}{2}}gg q^{4}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then for any <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>λ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>∖</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>0</m:mn> <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-2023-0470_eq_0267.png\" /> <jats:tex-math>{lambdainmathbb{F}_{q}setminus{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the number of congruence classes of triangles with vertices in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:mo>×</m:mo> <m:mi>B</m:mi> <m:mo>×</m:mo> <m:mi>C</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0470_eq_0174.png\" /> <jats:tex-math>{Atimes Btimes C}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and one side-l","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299684","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 concerns the existence and multiplicity of solutions for a nonlinear Schrödinger–Kirchhoff-type equation involving the fractional p-Laplace operator in ℝN{mathbb{R}^{N}}. Precisely, we study the Kirchhoff-type problem (a+b∬ℝ2N|u(x)-u(y)|p|x-y|N+spdxdy)(-Δ)psu+V(x)|u|p-2u=f(x,u)
本文涉及ℝ N {mathbb{R}^{N} 中涉及分数 p-Laplace 算子的非线性薛定谔-基尔霍夫（Schrödinger-Kirchhoff）型方程的解的存在性和多重性。 .确切地说，我们研究的是基尔霍夫型问题 ( a + b ∵ ℝ 2 N | u ( x ) - u ( y ) | p | x - y | N + s p d x d y ) 。 ( - Δ ) p s u + V ( x ) | u | p - 2 u = f ( x , u ) in ℝ N , Biggl{(}a+biint_{mathbb{R}^{2N}}frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}、% mathrm{d}x,mathrm{d}yBiggr{)}(-Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u)quad% text{in }mathbb{R}^{N}, where a , b >；0 {a,b>0} , ( - Δ ) p s {(-Delta)^{s}_{p}} 是分数 p 拉普拉卡方，0 < s < 1 < p < N s {0<s<1<p<frac{N}{s}} V : ℝ N → ℝ {Vcolonmathbb{R}^{N}tomathbb{R}} 和 f : ℝ N × ℝ → ℝ {fcolonmathbb{R}^{N}timesmathbb{R}} 是连续函数，而 V 可以有负值，f 满足适当的增长假设。根据电势在无穷远处的衰减与非线性项在原点的行为之间的相互作用，利用惩罚论证以及 L ∞ {L^{infty}} -估计和变分法，我们证明了正解的存在。此外，只要非线性项为奇数，我们还证明了无穷多个解的存在。
In this paper, we investigate nonlocal partial systems that incorporate the fractional Laplace operator. Our primary focus is to establish a theorem concerning the existence of optimal solutions for these equations. To achieve this, we utilize two fundamental tools: information obtained from an iterative reconstruction algorithm and a variant of the Phragmén–Lindelöf principle of concentration and compactness tailored for fractional systems. By employing these tools, we provide valuable insights into the nature of nonlocal partial systems and their optimal solutions.
{"title":"The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth","authors":"Siyao Peng","doi":"10.1515/forum-2023-0265","DOIUrl":"https://doi.org/10.1515/forum-2023-0265","url":null,"abstract":"In this paper, we investigate nonlocal partial systems that incorporate the fractional Laplace operator. Our primary focus is to establish a theorem concerning the existence of optimal solutions for these equations. To achieve this, we utilize two fundamental tools: information obtained from an iterative reconstruction algorithm and a variant of the Phragmén–Lindelöf principle of concentration and compactness tailored for fractional systems. By employing these tools, we provide valuable insights into the nature of nonlocal partial systems and their optimal solutions.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300114","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}
Given a totally real algebraic number field k of degree s, we consider locally symmetric spaces XG/Γ{X_{G}/Gamma} associated with arithmetic subgroups Γ of the special linear algebraic k-group G=SLM2(D){G=mathrm{SL}_{M_{2}(D)}}, attached to a quaternion division k-algebra D. The group G is k-simple, of k-rank one, and non-split over k. Using reduction theory, one can construct an open subset YΓ⊂XG/Γ{Y_{Gamma}subset X_{G}/Gamma} such that its closure Y¯Γ{overline{Y}_{Gamma}} is a compact manifold with boundary ∂Y¯Γ
给定阶数为 s 的全实代数数域 k，我们考虑与特殊线性代数 k 群 G = SL M 2 ( D ) {G=mathrm{SL}_{M_{2}(D)}} 的算术子群 Γ 相关联的局部对称空间 X G /Γ {X_{G}/Gamma} 。 群 G 是 k 简单的、k秩为 1 的、对 k 非分裂的。利用还原理论，我们可以构造一个开放子集 Y Γ ⊂ X G / Γ {Y_{Gamma}subset X_{G}//{Gamma}，使得它的闭包 Y ¯ Γ {overline{Y}_{Gamma} 是一个边界为 ∂ Y ¯ Γ {partialoverline{Y}_{Gamma} 的紧凑流形。｝ 包含 Y ¯ Γ → X G / Γ {overline{Y}_{Gamma}}/rightarrow X_{G}/Gamma} 是同调等价的。边界 ∂ Y ¯ Γ {partialoverline{Y}_{Gamma} 的连通分量 Y [ P ] {Y^{[P]}} 与 G 的最小抛物 k 子群的有限集合 Γ 共轭类一一对应。首先，如果四元除法 k 代数 D 是全定的，即 D 在 k 的所有阿基米德位置上都是斜的，那么我们证明这个束的基础与维数为 s - 1 {s-1} 的环 T s - 1 {T^{s-1} 是同构的，具有紧凑纤维 T 4 s {T^{4s}} ，其结构群是 SL 4 s {T^{4s}} 。 其结构群为 SL 4 s ( ℤ ) {mathrm{SL}_{4s}(mathbb{Z})} 。我们确定了 Y [ P ] {Y^{[P]}} 的同调。其次，如果四元数除 k 代数 D 是不确定的，那么至少存在一个拱顶位置 v∈V k , ∞ {vin V_{k,infty}} ，在这个位置上 D v {D_{v}} 分裂于 ℝ {mathbb{R}} 。 即 D v ≅ M 2 ( ℝ ) {D_{v}cong M_{2}(mathbb{R})} ，纤维是同构的。 ，纤维与 T 4 s {T^{4s}} 同构。 但是束的基空间更为复杂。
{"title":"On arithmetic quotients of the group SL2 over a quaternion division k-algebra","authors":"Sophie Koch, Joachim Schwermer","doi":"10.1515/forum-2023-0422","DOIUrl":"https://doi.org/10.1515/forum-2023-0422","url":null,"abstract":"Given a totally real algebraic number field <jats:italic>k</jats:italic> of degree <jats:italic>s</jats:italic>, we consider locally symmetric spaces <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0422_eq_0351.png\" /> <jats:tex-math>{X_{G}/Gamma}</jats:tex-math> </jats:alternatives> </jats:inline-formula> associated with arithmetic subgroups Γ of the special linear algebraic <jats:italic>k</jats:italic>-group <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:msub> <m:mi>SL</m:mi> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>D</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0422_eq_0183.png\" /> <jats:tex-math>{G=mathrm{SL}_{M_{2}(D)}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, attached to a quaternion division <jats:italic>k</jats:italic>-algebra <jats:italic>D</jats:italic>. The group <jats:italic>G</jats:italic> is <jats:italic>k</jats:italic>-simple, of <jats:italic>k</jats:italic>-rank one, and non-split over <jats:italic>k</jats:italic>. Using reduction theory, one can construct an open subset <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>Y</m:mi> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:msub> <m:mo>⊂</m:mo> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0422_eq_0361.png\" /> <jats:tex-math>{Y_{Gamma}subset X_{G}/Gamma}</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that its closure <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mover accent=\"true\"> <m:mi>Y</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0422_eq_0583.png\" /> <jats:tex-math>{overline{Y}_{Gamma}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a compact manifold with boundary <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∂</m:mo> <m:mo></m:mo> <m:msub> <m:mover accent=\"true\"> <m:mi>Y</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xli","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299886","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 several versions of the classical Paley inequality for the Weyl transform. As for some applications, we prove a version of the Hörmander’s multiplier theorem to discuss Lp{L^{p}}-Lq{L^{q}} boundedness of the Weyl multipliers and prove the Hardy–Littlewood inequality. We also consider the vector-valued version of the inequalities of Paley, Hausdorff–Young, and Hardy–Littlewood and their relations. Finally, we also prove Pitt’s inequality for the Weyl transform.
在本文中，我们证明了韦尔变换经典帕利不等式的几个版本。至于一些应用，我们证明了赫曼德乘数定理的一个版本，讨论了韦尔乘数的 L p {L^{p}} L q {L^{q}} 有界性，并证明了哈代-利特尔伍德不等式。 - L q {L^{q}} 的有界性，并证明了哈代-利特尔伍德不等式。我们还考虑了 Paley、Hausdorff-Young 和 Hardy-Littlewood 不等式的向量值版本及其关系。最后，我们还证明了韦尔变换的皮特不等式。
{"title":"Paley inequality for the Weyl transform and its applications","authors":"Ritika Singhal, N. Shravan Kumar","doi":"10.1515/forum-2023-0302","DOIUrl":"https://doi.org/10.1515/forum-2023-0302","url":null,"abstract":"In this paper, we prove several versions of the classical Paley inequality for the Weyl transform. As for some applications, we prove a version of the Hörmander’s multiplier theorem to discuss <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-2023-0302_eq_0237.png\" /> <jats:tex-math>{L^{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:msup> <m:mi>L</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0302_eq_0241.png\" /> <jats:tex-math>{L^{q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> boundedness of the Weyl multipliers and prove the Hardy–Littlewood inequality. We also consider the vector-valued version of the inequalities of Paley, Hausdorff–Young, and Hardy–Littlewood and their relations. Finally, we also prove Pitt’s inequality for the Weyl transform.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299847","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 A and B be two unipotent elements of SU(2,1){mathrm{SU}(2,1)} with distinct fixed points. In [S. B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 2023, 1, Paper No. 9], the authors gave several conditions that guarantee the subgroup 〈A,B〉{langle A,Brangle} is discrete and free by using Klein’s combination theorem. We will improve their conditions by using a variant of Klein’s combination theorem. With the same arguments and the additional assumption that AB is unipotent, we also extend Parker and Will’s condition that guarantees the subgroup 〈A,B〉{langle A,Brangle} is discrete and free in [J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 2017, 6, 3391–3451].
设 A 和 B 是 SU ( 2 , 1 ) {mathrm{SU}(2,1)} 的两个单能元，它们有不同的定点。在 [S. B. Kalane 和 J. R. Parker.B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 2023, 1.Z. 303 2023, 1, Paper No. 9]中，作者给出了几个条件，利用克莱因组合定理保证子群 〈 A , B 〉 {langle A,Brangle} 是离散和自由的。我们将利用克莱因组合定理的一个变体来改进它们的条件。通过同样的论证和 AB 是单能的这一额外假设，我们还扩展了帕克和威尔在 [J. R. Parker and P. Will] 中提出的保证子群 〈 A , B 〉 {langle A,Brangle} 是离散和自由的条件。R. Parker and P. Will, A complex hyperbolic Riley slice, Geom.Topol.21 2017, 6, 3391-3451].
{"title":"Free groups generated by two unipotent maps","authors":"Chao Jiang, Baohua Xie","doi":"10.1515/forum-2023-0442","DOIUrl":"https://doi.org/10.1515/forum-2023-0442","url":null,"abstract":"Let <jats:italic>A</jats:italic> and <jats:italic>B</jats:italic> be two unipotent elements of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>SU</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <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-2023-0442_eq_0338.png\" /> <jats:tex-math>{mathrm{SU}(2,1)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with distinct fixed points. In [S. B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 2023, 1, Paper No. 9], the authors gave several conditions that guarantee the subgroup <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">〈</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">〉</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0442_eq_0313.png\" /> <jats:tex-math>{langle A,Brangle}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is discrete and free by using Klein’s combination theorem. We will improve their conditions by using a variant of Klein’s combination theorem. With the same arguments and the additional assumption that <jats:italic>AB</jats:italic> is unipotent, we also extend Parker and Will’s condition that guarantees the subgroup <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">〈</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">〉</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0442_eq_0313.png\" /> <jats:tex-math>{langle A,Brangle}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is discrete and free in [J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 2017, 6, 3391–3451].","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300127","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}
Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.
{"title":"Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations","authors":"Monir Chadad, Mohamed Erraoui","doi":"10.1515/forum-2023-0288","DOIUrl":"https://doi.org/10.1515/forum-2023-0288","url":null,"abstract":"Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299674","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 one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form f and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the p-adic L-function defined by Magrone.
我们证明了满足 "宽松 "希格纳假设的高权重普通模形式 f 和虚二次域的反周岩泽主猜想的一个可分性关系。让Λ成为反周岩泽代数。按照霍华德（Howard）和隆戈-维尼（Longo-Vigni）的方法，我们从合适的志村曲线上的 Heegner 点出发，构建了广义 Heegner 类的Λ-adic Kolyvagin 系统。作为其应用，我们还证明了岩泽-格林伯格主猜想中关于马格隆定义的 p-adic L 函数的可分性关系。
{"title":"On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting","authors":"Maria Rosaria Pati","doi":"10.1515/forum-2023-0141","DOIUrl":"https://doi.org/10.1515/forum-2023-0141","url":null,"abstract":"We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form <jats:italic>f</jats:italic> and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the <jats:italic>p</jats:italic>-adic <jats:italic>L</jats:italic>-function defined by Magrone.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300117","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 deal with the existence of nontrivial nonnegative solutions for a (p,N){(p,N)}-Laplacian Schrödinger–Kirchhoff problem in ℝN{mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the (p,N){(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.
本文讨论了具有奇异指数非线性的ℝ N {mathbb{R}^{N}} 中 ( p , N ) {(p,N)} - 拉普拉契亚薛定谔-基尔霍夫问题的非微观非负解的存在性。本文的主要特点是椭圆算子的 ( p , N ) {(p,N)} 增长、双重不紧凑性以及基尔霍夫函数属于退化类型。为了建立存在性结果，我们使用了山口定理、埃克兰变分原理、奇异特鲁丁格-莫泽不等式，以及奇异指数非线性的全新布雷齐斯-利布型定理。
{"title":"Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N","authors":"Deepak Kumar Mahanta, Tuhina Mukherjee, Abhishek Sarkar","doi":"10.1515/forum-2023-0407","DOIUrl":"https://doi.org/10.1515/forum-2023-0407","url":null,"abstract":"In this paper, we deal with the existence of nontrivial nonnegative solutions for a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0407_eq_0313.png\" /> <jats:tex-math>{(p,N)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Laplacian Schrödinger–Kirchhoff problem in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</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-2023-0407_eq_0469.png\" /> <jats:tex-math>{mathbb{R}^{N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with singular exponential nonlinearity. The main features of the paper are the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0407_eq_0313.png\" /> <jats:tex-math>{(p,N)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300116","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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