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{"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}

{"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}

{"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>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo>&lt;</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&lt;pleq q&lt;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}

{"title":"The 𝐿𝑝 restriction bounds for Neumann data on surface","authors":"Xianchao Wu","doi":"10.1515/forum-2024-0237","DOIUrl":"https://doi.org/10.1515/forum-2024-0237","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;{&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mi&gt;λ&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mo stretchy=\"false\"&gt;}&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0237_ineq_0001.png\"/&gt; &lt;jats:tex-math&gt;{u_{lambda}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a sequence of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;L&lt;/m:mi&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0237_ineq_0002.png\"/&gt; &lt;jats:tex-math&gt;L^{2}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;M&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0237_ineq_0003.png\"/&gt; &lt;jats:tex-math&gt;(M,g)&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. We seek to get an &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;L&lt;/m:mi&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0237_ineq_0004.png\"/&gt; &lt;jats:tex-math&gt;L^{p}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; restriction bound of the Neumann data &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mi&gt;λ&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo lspace=\"0em\"&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mo rspace=\"0em\"&gt;∂&lt;/m:mo&gt; &lt;m:mi&gt;ν&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:msub&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mi&gt;λ&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mi&gt;γ&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0237_ineq_0005.png\"/&gt; &lt;jats:tex-math&gt;lambda^{-1}partial_{nu}u_{lambda}|_{gamma}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; along a unit geodesic 𝛾. Using the 𝑇-&lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;T&lt;/m:mi&gt; &lt;m:mo&gt;∗&lt;/m:mo&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0237_ineq_0006.png\"/&gt; &lt;jats:tex-math&gt;T^{*}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; argument, one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is &lt;jats:inline-formula&gt; &lt;jats:alter","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141586365","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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}