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":"21 1","pages":""},"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 <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
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":"32 1","pages":""},"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":"26 1","pages":""},"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":"35 1","pages":""},"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":"155 1","pages":""},"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":"21 1","pages":""},"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}
Attila Bérczes, Yann Bugeaud, Kálmán Győry, Jorge Mello, Alina Ostafe, Min Sha
Let <jats:italic>f</jats:italic> be a polynomial with coefficients in the ring <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>O</m:mi> <m:mi>S</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0381_eq_0397.png" /> <jats:tex-math>{O_{S}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:italic>S</jats:italic>-integers of a number field <jats:italic>K</jats:italic>, <jats:italic>b</jats:italic> a non-zero <jats:italic>S</jats:italic>-integer, and <jats:italic>m</jats:italic> an integer <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <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-2023-0381_eq_0497.png" /> <jats:tex-math>{geq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We consider the following equation <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:mo>⋆</m:mo> <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-2023-0381_eq_0302.png" /> <jats:tex-math>{(star)}</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:mrow> <m:mi>f</m:mi> <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:mo>=</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo></m:mo> <m:msup> <m:mi>y</m:mi> <m:mi>m</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0381_eq_0620.png" /> <jats:tex-math>{f(x)=by^{m}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi>O</m:mi> <m:mi>S</m:mi> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0381_eq_0734.png" /> <jats:tex-math>{x,yin O_{S}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of <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:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>f</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-2023-0381_eq_0344.png" /> <jats:tex-math>{K,S,f,m}</jats:tex-math> </jats:altern
设 f 是一个多项式，其系数在数域 K 的 S 整数环 O S {O_{S}} 中，b 是一个非零 S 整数，m 是一个整数 ≥ 2 {geq 2} 。我们考虑以下方程 ( ⋆ ) {(star)} : f ( x ) = b y m {f(x)=by^{m}} in x , y ∈ O S {x,yin O_{S}}. .在著名的 LeVeque 条件下，我们给出了方程 ( ⋆ ) {(star)} 的解 x 的高度的 K , S , f , m {K,S,f,m} 和 b 的 S-norm 的完全明确的上限。此外，我们还给出了 K , S , f {K,S,f} 和 b 的 S-norm 的明确约束 C，即如果 m > C {m>C} 等式 ( ⋆ ) {(star)} 只有 y = 0 {y=0} 或一个同根的解。我们的结果是 Trelina、Brindza、Shorey 和 Tijdeman、Voutier 和 Bugeaud 工作的更详细版本，并将 Bérczes、Evertse 和 Győry 早期的结果扩展到多根多项式。与之前的结果不同，我们的界限取决于 b 的 S-norm 而不是其高度。
{"title":"Explicit bounds for the solutions of superelliptic equations over number fields","authors":"Attila Bérczes, Yann Bugeaud, Kálmán Győry, Jorge Mello, Alina Ostafe, Min Sha","doi":"10.1515/forum-2023-0381","DOIUrl":"https://doi.org/10.1515/forum-2023-0381","url":null,"abstract":"Let <jats:italic>f</jats:italic> be a polynomial with coefficients in the ring <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>O</m:mi> <m:mi>S</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0381_eq_0397.png\" /> <jats:tex-math>{O_{S}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:italic>S</jats:italic>-integers of a number field <jats:italic>K</jats:italic>, <jats:italic>b</jats:italic> a non-zero <jats:italic>S</jats:italic>-integer, and <jats:italic>m</jats:italic> an integer <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <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-2023-0381_eq_0497.png\" /> <jats:tex-math>{geq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We consider the following equation <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:mo>⋆</m:mo> <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-2023-0381_eq_0302.png\" /> <jats:tex-math>{(star)}</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:mrow> <m:mi>f</m:mi> <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:mo>=</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo></m:mo> <m:msup> <m:mi>y</m:mi> <m:mi>m</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0381_eq_0620.png\" /> <jats:tex-math>{f(x)=by^{m}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi>O</m:mi> <m:mi>S</m:mi> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0381_eq_0734.png\" /> <jats:tex-math>{x,yin O_{S}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of <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:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>f</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-2023-0381_eq_0344.png\" /> <jats:tex-math>{K,S,f,m}</jats:tex-math> </jats:altern","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300125","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":"1 1","pages":""},"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}
In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM) bundles and Ulrich bundles on rational homogeneous spaces. From this result, we see that there are only finitely many irreducible homogeneous ACM bundles (up to twist) and Ulrich bundles on these varieties. Moreover, we give numerical criteria for some special irreducible homogeneous bundles to be ACM bundles.
{"title":"Homogeneous ACM and Ulrich bundles on rational homogeneous spaces","authors":"Xinyi Fang","doi":"10.1515/forum-2023-0383","DOIUrl":"https://doi.org/10.1515/forum-2023-0383","url":null,"abstract":"In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM) bundles and Ulrich bundles on rational homogeneous spaces. From this result, we see that there are only finitely many irreducible homogeneous ACM bundles (up to twist) and Ulrich bundles on these varieties. Moreover, we give numerical criteria for some special irreducible homogeneous bundles to be ACM bundles.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197511","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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