We prove a general result concerning properties preserved under certain amalgamated free products.
我们证明了一个关于在某些汞齐化自由乘积下保留的性质的一般结果。
{"title":"A note on almalgamated free products","authors":"Qihui Li, Don Hadwin, Junhao Shen","doi":"10.1090/proc/16791","DOIUrl":"https://doi.org/10.1090/proc/16791","url":null,"abstract":"<p>We prove a general result concerning properties preserved under certain amalgamated free products.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151831","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}
There is a long-standing problem, posed by A.T.-M. Lau [Fixed point theory and its applications, Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak∗^{ast } continuous nonexpansive semigroup action on a nonempty weak∗^{ast } compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak∗^{ast } compact convex sets with the Radon–Nikodým property.
A.T.-M. Lau [Fixed point theory and its applications, Academic Press, New York-London, 1976, pp.Lau [Fixed point theory and its applications, Academic Press, New York-London, 1976, pp.在本论文中，我们讨论了这一问题的现状，并给出了具有 Radon-Nikodým 性质的弱∗ ^{ast } 紧凑凸集的部分解决方案。
{"title":"Radon–Nikodým property and Lau’s conjecture","authors":"Andrzej Wiśnicki","doi":"10.1090/proc/16884","DOIUrl":"https://doi.org/10.1090/proc/16884","url":null,"abstract":"<p>There is a long-standing problem, posed by A.T.-M. Lau [<italic>Fixed point theory and its applications</italic>, Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Superscript asterisk\"> <mml:semantics> <mml:msup> <mml:mi/> <mml:mrow> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">^{ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> continuous nonexpansive semigroup action on a nonempty weak<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Superscript asterisk\"> <mml:semantics> <mml:msup> <mml:mi/> <mml:mrow> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">^{ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Superscript asterisk\"> <mml:semantics> <mml:msup> <mml:mi/> <mml:mrow> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">^{ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> compact convex sets with the Radon–Nikodým property.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872178","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}
Srivatsav Kunnawalkam Elayavalli, Koichi Oyakawa, Forte Shinko, Pieter Spaas
We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfiniteness of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.
{"title":"Hyperfiniteness for group actions on trees","authors":"Srivatsav Kunnawalkam Elayavalli, Koichi Oyakawa, Forte Shinko, Pieter Spaas","doi":"10.1090/proc/16851","DOIUrl":"https://doi.org/10.1090/proc/16851","url":null,"abstract":"<p>We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfiniteness of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744759","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 article, we show the necessary and sufficient conditions for the inequality ‖u‖LtqLxr≲‖u‖Xs,b,begin{equation*} |u|_{L_t^qL_x^r}lesssim |u|_{X^{s,b}}, end{equation*} where ‖u‖Xs,b≔‖u^(τ,ξ)⟨ξ⟩s⟨τ+|ξ|
在本文中，我们展示了不等式 ‖ u ‖ L t q L x r ≲ ‖ u ‖ X s , b 的必要条件和充分条件。|u|_{L_t^qL_x^r}lesssim |u|_{X^{s,b}}, end{equation*} 其中 ‖ u ‖ X s , b ≔ ‖ u ^ ( τ , ξ ) ξ ⟨ s τ + | ξ | 2 ⟩ b ‖ L τ 、ξ 2 ||u_{X^{s,b}}≔||hat{u}(tau ,xi)（矩形 xi ）（矩形 tau + |xi |^2rangle ^b |{L_{tau,xi}^2}。这些估计也被称为与薛定谔方程有关的斯特里查兹估计。我们还给出了薛定谔方程和艾里方程解的最大函数估计的新证明。
{"title":"Strichartz type estimates for solutions to the Schrödinger equation","authors":"Jie Chen","doi":"10.1090/proc/16887","DOIUrl":"https://doi.org/10.1090/proc/16887","url":null,"abstract":"<p>In this article, we show the necessary and sufficient conditions for the inequality <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-vertical-bar u double-vertical-bar Subscript upper L Sub Subscript t Sub Superscript q Subscript upper L Sub Subscript x Sub Superscript r Subscript Baseline less-than-or-equivalent-to double-vertical-bar u double-vertical-bar Subscript upper X Sub Superscript s comma b Subscript Baseline comma\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">‖</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">‖</mml:mo> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>t</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mi>r</mml:mi> </mml:msubsup> </mml:mrow> </mml:msub> <mml:mo>≲</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">‖</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">‖</mml:mo> <mml:mrow> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">begin{equation*} |u|_{L_t^qL_x^r}lesssim |u|_{X^{s,b}}, end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-vertical-bar u double-vertical-bar Subscript upper X Sub Superscript s comma b Baseline colon-equal double-vertical-bar ModifyingAbove u With caret left-parenthesis tau comma xi right-parenthesis mathematical left-angle xi mathematical right-angle Superscript s Baseline mathematical left-angle tau plus StartAbsoluteValue xi EndAbsoluteValue squared mathematical right-angle Superscript b Baseline double-vertical-bar Subscript upper L Sub Subscript tau comma xi Sub Superscript 2\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">‖</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">‖</mml:mo> <mml:mrow> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>≔</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">‖</mml:mo> <mml:mrow> <mml:mover> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">^</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo> <mml:mi>ξ</mml:mi> <mml:msup> <mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo> <mml:mi>s</mml:mi> </mml:msup> <mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>ξ</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <m","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744657","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}
Consider L0L^0, the FF-space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin’s classical result on the factorization of sublinear continuous operators from a Banach space to L0L^0, we prove a theorem that characterizes those maps from any quasi-metric space into L0L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces.
L 0 L^0 是有限度量空间上可测函数的所有等价类的 F F 空间，具有度量收敛拓扑。受尼基申关于从巴纳赫空间到 L 0 L^0 的亚线性连续算子因子化的经典结果的启发，我们证明了一个定理，它描述了从任何准度量空间到 L 0 L^0 的映射的特征，这些映射通过 Marcinkiewicz 加权空间强因子化。我们展示了对某类由奥利奇序列空间生成的广义拉德马赫型准巴纳赫空间上的亚线性算子的应用。
{"title":"Nikishin’s theorem and factorization through Marcinkiewicz spaces","authors":"Mieczysław Mastyło, Enrique Sánchez Pérez","doi":"10.1090/proc/16888","DOIUrl":"https://doi.org/10.1090/proc/16888","url":null,"abstract":"<p>Consider <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 0\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=\"application/x-tex\">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin’s classical result on the factorization of sublinear continuous operators from a Banach space to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 0\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we prove a theorem that characterizes those maps from any quasi-metric space into <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 0\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945161","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 1990 F. Chiarenza and M. Frasca published a paper in which they generalized a result of C. Fefferman on estimates of the integral of |bu|p|bu|^{p} through the integral of |Du|p|Du|^{p} for p>1p>1. Formally their proof is valid only for d≥3dgeq 3. We present here further generalization with a different proof in which DD is replaced with the fractional power of the Laplacian for any dimension d≥2dgeq 2.
1990 年，F. Chiarenza 和 M. Frasca 发表论文，将 C. Fefferman 关于 p > 1 p>1 时通过 | D u | p |Du|^{p} 的积分来估计 | b u | p |bu|^{p} 的积分的结果加以推广。形式上，他们的证明仅对 d ≥ 3 dgeq 3 有效。我们在这里用一个不同的证明来进一步概括，在这个证明中，对于任何维度 d ≥ 2 dgeq 2，D D 被替换为拉普拉奇的分数幂。
{"title":"A remark on a paper by F. Chiarenza and M. Frasca","authors":"N. Krylov","doi":"10.1090/proc/16885","DOIUrl":"https://doi.org/10.1090/proc/16885","url":null,"abstract":"<p>In 1990 F. Chiarenza and M. Frasca published a paper in which they generalized a result of C. Fefferman on estimates of the integral of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue b u EndAbsoluteValue Superscript p\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>b</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|bu|^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> through the integral of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue upper D u EndAbsoluteValue Superscript p\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|Du|^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Formally their proof is valid only for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dgeq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We present here further generalization with a different proof in which <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is replaced with the fractional power of the Laplacian for any dimension <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 2\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dgeq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197124","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}
A celebrated result in probability theory is that a simple symmetric random walk on the dd-dimensional lattice Zdmathbb {Z}^d is recurrent for d=1,2d=1,2 and transient for d≥3dgeq 3. In this note, we derive a closed-form expression, in terms of the Lauricella function FCF_C, for the return probability for all d≥3dgeq 3. Previously, a closed-form formula had only been available for d=3d=3.
概率论中一个著名的结果是，在 d d 维网格 Z d mathbb {Z}^d 上的简单对称随机行走在 d = 1 , 2 d=1,2 时是经常性的，而在 d ≥ 3 dgeq 3 时是瞬时性的。在本说明中，我们用劳里切拉函数 F C F_C 为所有 d ≥ 3 dgeq 3 的回归概率推导出一个闭式表达式。在此之前，只有 d=3 d=3 时才有闭式公式。
{"title":"On Pólya’s random walk constants","authors":"Robert Gaunt, Saralees Nadarajah, Tibor Pogány","doi":"10.1090/proc/16854","DOIUrl":"https://doi.org/10.1090/proc/16854","url":null,"abstract":"<p>A celebrated result in probability theory is that a simple symmetric random walk on the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=\"application/x-tex\">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional lattice <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Superscript d\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is recurrent for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d equals 1 comma 2\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d=1,2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and transient for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dgeq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this note, we derive a closed-form expression, in terms of the Lauricella function <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F Subscript upper C\"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">F_C</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for the return probability for all <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dgeq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Previously, a closed-form formula had only been available for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d equals 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503309","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 describe the intersection of the Torelli locus j(M4ct)=J4j(mathcal {M}_4^{ct}) = mathcal {J}_4 with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic 2. We show that the locus of supersingular Jacobians S4∩J4mathcal {S}_4cap mathcal {J}_4 in characteristic 2 is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over F2mathbb {F}_2, and another involves a more careful study of some relevant Ekedahl-Oort loci.
我们描述了与特征 2 中的超星点相关的牛顿和埃克达尔-奥尔特地层与托雷利点 j ( M 4 c t ) = J 4 j(mathcal {M}_4^{ct}) = mathcal {J}_4 的交集。我们证明了在特征 2 中超共轭雅各布数 S 4 ∩ J 4 mathcal {S}_4cap mathcal {J}_4 的位置是纯三维的。要得到这个结果，一种方法是分析定义在 F 2 mathbb {F}_2 上的光滑四属曲线和主要极化无边四褶的数据，另一种方法是对一些相关的埃克达尔-奥尔特（Ekedahl-Oort）位点进行更细致的研究。
{"title":"Supersingular curves of genus four in characteristic two","authors":"Dušan Dragutinović","doi":"10.1090/proc/16792","DOIUrl":"https://doi.org/10.1090/proc/16792","url":null,"abstract":"<p>We describe the intersection of the Torelli locus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"j left-parenthesis script upper M 4 Superscript c t Baseline right-parenthesis equals script upper J 4\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>j</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msubsup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>4</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msubsup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">J</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">j(mathcal {M}_4^{ct}) = mathcal {J}_4</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic 2. We show that the locus of supersingular Jacobians <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper S 4 intersection script upper J 4\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>∩<!-- ∩ --></mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">J</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {S}_4cap mathcal {J}_4</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in characteristic 2 is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper F 2\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {F}_2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and another involves a more careful study of some relevant Ekedahl-Oort loci.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140712543","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 show that a transversely geometrically controlling boundary damping strip is sufficient but not necessary for t−1/2t^{-1/2}-decay of waves on product manifolds. We give a general scheme to turn resolvent estimates for impedance problems on cross-sections to wave decay on product manifolds.
我们证明了横向几何控制边界阻尼条对于乘积流形上波的 t - 1 / 2 t^{-1/2} 衰减是足够的，而不是必要的。我们给出了将横截面上阻抗问题的解析估计转化为积流形上波衰减的一般方案。
{"title":"Stabilisation of waves on product manifolds by boundary strips","authors":"Ruoyu Wang","doi":"10.1090/proc/16242","DOIUrl":"https://doi.org/10.1090/proc/16242","url":null,"abstract":"<p>We show that a transversely geometrically controlling boundary damping strip is sufficient but not necessary for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t Superscript negative 1 slash 2\"> <mml:semantics> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">t^{-1/2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-decay of waves on product manifolds. We give a general scheme to turn resolvent estimates for impedance problems on cross-sections to wave decay on product manifolds.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we prove a family of sharp weighed inequalities which involve weighted kk-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in Rnmathbb {R}^n. This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao’s results.
在本注释中，我们证明了一系列尖锐的权重不等式，它们涉及 R n mathbb {R}^n 中封闭超曲面的加权 k k -th 平均曲率积分和两个不同的质点积分。这个不等式概括了 Wei 和 Zhou [Bull. Lond. Math. Soc. 55 (2023), pp.267 (2014), pp.Contemp.Math.17 (2015), p. 1550014].这里我们提出一个不依赖邝淼结果的证明。
{"title":"A note on new weighted geometric inequalities for hypersurfaces in ℝⁿ","authors":"Jie Wu","doi":"10.1090/proc/16875","DOIUrl":"https://doi.org/10.1090/proc/16875","url":null,"abstract":"<p>In this note, we prove a family of sharp weighed inequalities which involve weighted <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript n\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao’s results.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744655","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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