本文研究了以下具有组合非线性的分数薛定谔方程的归一化解 { ( - Δ ) s u = λ u + μ | u | q - 2 u + | u | p - 2 u in ℝ N , ∫ ℝ N u 2 𝑑 x = a 2 , displaystyleleft{begin{aligned}(-Delta)^{s}u&% displaystyle=lambda u+mulvert urvert^{q-2}u+lvert urvert^{p-2}u&&；% displaystylephantom{}text{in }mathbb{R}^{N},displaystyleint_{mathbb{R}^{N}}u^{2},dx&displaystyle=a^{2},end{aligned}right. 其中，0 < s < 1 {0<s<1} , N > 2 s {N>2s} , 2 < q < 1 {0<s<1}. 2 < q < p = 2 s * = 2 N N - 2 s {2<q<p=2_{s}^{*}=frac{2N}{N-2s}} , a , μ > 0 {a,mu>0} 且 λ∈ ℝ {lambdainmathbb{R}} 是拉格朗日乘数。由于 p< 2 s * {p<2_{s}^{*}}的存在性结果已被证明，因此使用近似法，即让 p → 2 s * {prightarrow 2_{s}^{*}} ，我们可以得到几个存在性结果。 ，我们得到了几个存在性结果。此外，我们还分析了当μ → 0 {murightarrow 0}和μ达到其上限时解的渐近行为。
In this article, we study the extent to which an n-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an n-dimensional compact flat manifold with the cyclic holonomy group of square-free order.
本文研究了具有无平方阶循环全局群的 n 维紧凑平面流形在多大程度上可以通过其基群的有限商来区分。特别是，我们展示了一个具有无平方阶循环全局群的 n 维紧凑平面流形的基群无穷属的心数公式。
{"title":"Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order","authors":"Genildo de Jesus Nery","doi":"10.1515/forum-2021-0298","DOIUrl":"https://doi.org/10.1515/forum-2021-0298","url":null,"abstract":"In this article, we study the extent to which an <jats:italic>n</jats:italic>-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an <jats:italic>n</jats:italic>-dimensional compact flat manifold with the cyclic holonomy group of square-free order.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656232","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 Boidol group is the smallest non-∗{ast}-regular exponential Lie group. It is of dimension 4 and its Lie algebra is an extension of the Heisenberg Lie algebra by the reals with the roots 1 and -1. We describe the C*-algebra of the Boidol group as an algebra of operator fields defined over the spectrum of the group. It is the only connected solvable Lie group of dimension less than or equal to 4 whose group C*-algebra had not yet been determined.
{"title":"The C*-algebra of the Boidol group","authors":"Ying-Fen Lin, Jean Ludwig","doi":"10.1515/forum-2021-0209","DOIUrl":"https://doi.org/10.1515/forum-2021-0209","url":null,"abstract":"The Boidol group is the smallest non-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mo>∗</m:mo> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2021-0209_eq_0575.png\" /> <jats:tex-math>{ast}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-regular exponential Lie group. It is of dimension 4 and its Lie algebra is an extension of the Heisenberg Lie algebra by the reals with the roots 1 and -1. We describe the C*-algebra of the Boidol group as an algebra of operator fields defined over the spectrum of the group. It is the only connected solvable Lie group of dimension less than or equal to 4 whose group C*-algebra had not yet been determined.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"291 2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656067","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 modern study of the exponential sums is mainly about their analytic estimates as complex numbers, which is local. In this paper, we study one global property of the exponential sums by viewing them as algebraic integers. For a kind of generalized Kloosterman sums, we present their degrees as algebraic integers.
{"title":"Degrees of generalized Kloosterman sums","authors":"Liping Yang","doi":"10.1515/forum-2023-0295","DOIUrl":"https://doi.org/10.1515/forum-2023-0295","url":null,"abstract":"The modern study of the exponential sums is mainly about their analytic estimates as complex numbers, which is local. In this paper, we study one global property of the exponential sums by viewing them as algebraic integers. For a kind of generalized Kloosterman sums, we present their degrees as algebraic integers.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"258 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645379","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 consider a Mikusiński-type convolution algebra Cα{C_{alpha}}, including functions with power-type singularities at the origin as well as all functions continuous on [0,∞){[0,infty)}. Algebraic properties of this space are derived, including its ideal structure, filtered and graded structure, and Jacobson radical. Applications to operators of fractional calculus and the associated integro-differential equations are discussed.
{"title":"Algebraic results on rngs of singular functions","authors":"Arran Fernandez, Müge Saadetoğlu","doi":"10.1515/forum-2023-0445","DOIUrl":"https://doi.org/10.1515/forum-2023-0445","url":null,"abstract":"We consider a Mikusiński-type convolution algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>C</m:mi> <m:mi>α</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0445_eq_0144.png\" /> <jats:tex-math>{C_{alpha}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, including functions with power-type singularities at the origin as well as all functions continuous on <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:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">∞</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-2023-0445_eq_0198.png\" /> <jats:tex-math>{[0,infty)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Algebraic properties of this space are derived, including its ideal structure, filtered and graded structure, and Jacobson radical. Applications to operators of fractional calculus and the associated integro-differential equations are discussed.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"37 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645371","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 establish the sharp Calderón–Vaillancourt theorem on LpL^{p} spaces for bi-parameter and bilinear pseudo-differential operators with symbols of critical order by deriving a sufficient and necessary condition on its symbol. This sharpens the result of [G. Lu and L. Zhang, Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order, Forum Math. 28 2016, 6, 1087–1094] which was only proved for symbols of subcritical order.
本文通过推导符号上的充分必要条件，建立了 L p L^{p} 空间上具有临界阶符号的双参数和双线性伪微分算子的尖锐卡尔德隆-韦朗库尔定理。这使[G. Lu and L. Zhang, Bi-parameter and bilinear Calderón-Vaillancourt theorem with subcritical order, Forum Math.28 2016, 6, 1087-1094]，该结果只证明了亚临界阶的符号。
{"title":"Bi-parameter and bilinear Calderón–Vaillancourt theorem with critical order","authors":"Jiao Chen, Liang Huang, Guozhen Lu","doi":"10.1515/forum-2023-0458","DOIUrl":"https://doi.org/10.1515/forum-2023-0458","url":null,"abstract":"In this paper, we establish the sharp Calderón–Vaillancourt theorem 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/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0458_eq_0174.png\" /> <jats:tex-math>L^{p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> spaces for bi-parameter and bilinear pseudo-differential operators with symbols of critical order by deriving a sufficient and necessary condition on its symbol. This sharpens the result of [G. Lu and L. Zhang, Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order, Forum Math. 28 2016, 6, 1087–1094] which was only proved for symbols of subcritical order.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"177 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645367","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}
Michael Ruzhansky, Anjali Shriwastawa, Bankteshwar Tiwari
In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0319_eq_0172.png" /> <jats:tex-math>{p=1}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <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-2023-0319_eq_0087.png" /> <jats:tex-math>{1leq q<infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case <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-2023-0319_eq_0086.png" /> <jats:tex-math>{1<pleq q<infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The case <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0319_eq_0172.png" /> <jats:tex-math>{p=1}</jats:tex-math> </jats:alternatives> </jats:inline-formula> requires a different argument and does not follow as the limit of known inequalities for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0319_eq_0173.png" /> <jats:tex-math>{p>1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan–Hadamard manifolds for the case <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0319_eq_0172.png" /> <jats:tex-math>{p=1}
在本文中，我们研究了具有极性分解的度量空间上的两重哈代不等式，即 p = 1 {p=1} 和 1 ≤ q < ∞ {1leq q<infty} 的情况。这一结果是对 [M. Ruzhansky 和 D. Ruzhansky] 中得到的哈代不等式的补充。Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc.Roy.Soc. A. 475 2019, 2223, Article ID 20180310] 中 1 < p ≤ q < ∞ {1<pleq q<infty} 的情况。p = 1 {p=1}的情况需要不同的论证，并不是 p > 1 {p>1}的已知不等式的极限。作为副产品，我们还得到了既定不等式中的最佳常数。我们举例说明了在 p = 1 {p=1} 和 1 ≤ q < ∞ {1leq q<infty} 的情况下，同质李群、双曲空间和 Cartan-Hadamard 流形上新的加权哈代不等式。
{"title":"Hardy inequalities on metric measure spaces, IV: The case p=1","authors":"Michael Ruzhansky, Anjali Shriwastawa, Bankteshwar Tiwari","doi":"10.1515/forum-2023-0319","DOIUrl":"https://doi.org/10.1515/forum-2023-0319","url":null,"abstract":"In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0172.png\" /> <jats:tex-math>{p=1}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <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-2023-0319_eq_0087.png\" /> <jats:tex-math>{1leq q<infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case <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-2023-0319_eq_0086.png\" /> <jats:tex-math>{1<pleq q<infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The case <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0172.png\" /> <jats:tex-math>{p=1}</jats:tex-math> </jats:alternatives> </jats:inline-formula> requires a different argument and does not follow as the limit of known inequalities for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0173.png\" /> <jats:tex-math>{p>1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan–Hadamard manifolds for the case <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0172.png\" /> <jats:tex-math>{p=1}","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139471173","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 give some new results on transcendence on algebraic groups. These results extend some previous ones established on commutative or linear algebraic groups to arbitrary algebraic groups in complex and p-adic fields, respectively.
{"title":"Transcendence on algebraic groups","authors":"Duc Hiep Pham","doi":"10.1515/forum-2023-0078","DOIUrl":"https://doi.org/10.1515/forum-2023-0078","url":null,"abstract":"In this paper, we give some new results on transcendence on algebraic groups. These results extend some previous ones established on commutative or linear algebraic groups to arbitrary algebraic groups in complex and <jats:italic>p</jats:italic>-adic fields, respectively.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139421105","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 make explicit Bombieri’s refinement of Gallagher’s log-free “large sieve density estimate near σ=1{sigma=1}” for Dirichlet L-functions. We use this estimate and recent work of Green to prove that if N≥2{Ngeq 2} is an integer, A⊆{1,…,N}{Asubseteq{1,ldots,N}}, and for all primes p no two elements in A differ by p-1{p-1}, then |A|≪N1-10-18{|A|ll N^{1-10^{-18}}}. This strengthens a theorem of Sárközy.
我们明确了 Bombieri 对 Gallagher 的无对数 "σ = 1 {sigma=1} 附近的大筛密度估计 "的改进。 的 "大筛密度估计"。我们利用这一估计和格林的最新研究成果证明，如果 N ≥ 2 {Ngeq 2} 是整数，则 A ⊆ { 1 , ... , N } {Asubseteq{1,ldots,N}} 对于所有素数 p，A 中没有两个元素相差 p - 1 {p-1} ，那么 | A | ≪ N 1 - 10 - 18 {|A|ll N^{1-10^{-18}}} 。这加强了萨尔科齐的一个定理。
{"title":"An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes","authors":"Jesse Thorner, Asif Zaman","doi":"10.1515/forum-2023-0091","DOIUrl":"https://doi.org/10.1515/forum-2023-0091","url":null,"abstract":"We make explicit Bombieri’s refinement of Gallagher’s log-free “large sieve density estimate near <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>σ</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0510.png\" /> <jats:tex-math>{sigma=1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>” for Dirichlet <jats:italic>L</jats:italic>-functions. We use this estimate and recent work of Green to prove that if <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-2023-0091_eq_0355.png\" /> <jats:tex-math>{Ngeq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is an integer, <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:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0326.png\" /> <jats:tex-math>{Asubseteq{1,ldots,N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and for all primes <jats:italic>p</jats:italic> no two elements in <jats:italic>A</jats:italic> differ by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0579.png\" /> <jats:tex-math>{p-1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <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:msup> <m:mi>N</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:msup> <m:mn>10</m:mn> <m:mrow> <m:mo>-</m:mo> <m:mn>18</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0630.png\" /> <jats:tex-math>{|A|ll N^{1-10^{-18}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This strengthens a theorem of Sárközy.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"52 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139422645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
在本文中，我们首先关注以下分数 p-Laplace 问题的多重归一化解的存在性：{ ( - Δ ) p s v + 𝒱 ( ξ x ) | v | p - 2 v = λ | v | p - 2 v + f ( v ) in ℝ N , ∫ ℝ N | v | p 𝑑 x = a p , left{begin{aligned}(-Delta)_{p}^{s}v+mathcal{V}(xi x)% lvert vrvert^{p-2}v&；displaystyle=lambdalvert vrvert^{p-2}v+f(v)quad% text{in }mathbb{R}^{N},displaystyleint_{mathbb{R}^{N}}lvert vrvert^{p},dx&displaystyle=a^{p},%end{aligned}right. 其中 a , ξ > 0 {a,xi>0} p ≥ 2 {pgeq 2} , λ ∈ ℝ {lambdainmathbb{R}} 是作为拉格朗日乘数出现的未知参数，𝒱 : ℝ N → [ 0 , ∞ ) {mathcal{V}:mathbb{R}^{N}to[0,infty)}是一个连续函数，并且 f 是一个具有 L p {L^{p}} 的连续函数。 -次临界增长的连续函数。我们利用 Lusternik-Schnirelmann 范畴证明存在解的多重性。接下来，我们证明归一化解的数量至少是𝒱 {mathcal{V}} 的全局最小点的数量。 ，因为通过埃克兰德变分原理，ξ足够小。
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