Pub Date : 2024-05-28DOI: 10.1016/j.matpur.2024.05.002
Alexandru Aleman , Carme Cascante , Joan Fàbrega , Daniel Pascuas , José Ángel Peláez
For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by , , and . We are concerned with the study of the boundedness of operators in the algebra generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in are finite linear combinations of finite products (words) of which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of can be characterized by one of the conditions , or belongs to or the Bloch space, for some integer . However, it is also proved that there are many operators, even single words in whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in in terms of a “fra
{"title":"Words of analytic paraproducts on Hardy and weighted Bergman spaces","authors":"Alexandru Aleman , Carme Cascante , Joan Fàbrega , Daniel Pascuas , José Ángel Peláez","doi":"10.1016/j.matpur.2024.05.002","DOIUrl":"10.1016/j.matpur.2024.05.002","url":null,"abstract":"<div><p>For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>z</mi></mrow></msubsup><mi>f</mi><mo>(</mo><mi>ζ</mi><mo>)</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>ζ</mi><mo>)</mo><mi>d</mi><mi>ζ</mi></math></span>, <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>z</mi></mrow></msubsup><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>ζ</mi><mo>)</mo><mi>g</mi><mo>(</mo><mi>ζ</mi><mo>)</mo><mi>d</mi><mi>ζ</mi></math></span>, and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>z</mi><mo>)</mo><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></math></span>. We are concerned with the study of the boundedness of operators in the algebra <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> are finite linear combinations of finite products (words) of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> which may involve a large amount of cancellations to be understood. The results in <span>[1]</span> show that boundedness of operators in a fairly large subclass of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> can be characterized by one of the conditions <span><math><mi>g</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>, or <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> belongs to <span><math><mi>B</mi><mi>M</mi><mi>O</mi><mi>A</mi></math></span> or the Bloch space, for some integer <span><math><mi>n</mi><mo>></mo><mn>0</mn></math></span>. However, it is also proved that there are many operators, even single words in <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> in terms of a “fra","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000497/pdfft?md5=2e2fa7bc18dbfdcfb9a2877bf0d1a207&pid=1-s2.0-S0021782424000497-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-28DOI: 10.1016/j.matpur.2024.05.011
Juliusz Banecki
We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic to a regular one. Furthermore we prove that algebraic homotopy classes of spheres form a subgroup of the homotopy group, and that a similar result holds also for cohomotopy groups of arbitrary varieties.
{"title":"Algebraic homotopy classes","authors":"Juliusz Banecki","doi":"10.1016/j.matpur.2024.05.011","DOIUrl":"10.1016/j.matpur.2024.05.011","url":null,"abstract":"<div><p>We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic to a regular one. Furthermore we prove that algebraic homotopy classes of spheres form a subgroup of the homotopy group, and that a similar result holds also for cohomotopy groups of arbitrary varieties.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000588/pdfft?md5=db876b33b1eccaf152b3741e38a53afa&pid=1-s2.0-S0021782424000588-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-06DOI: 10.1016/j.matpur.2024.04.008
Kari Astala , Martí Prats , Eero Saksman
We study quasiconformal mappings in planar domains Ω and their regularity properties described in terms of Sobolev, Bessel potential or Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the geometry of the boundary ∂Ω and of the smoothness of the Beltrami coefficient, that guarantee the global regularity of the mappings in these classes. In the Triebel-Lizorkin class with smoothness below 1, the same conditions give global regularity in Ω for the principal solutions with Beltrami coefficient supported in Ω.
{"title":"Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale","authors":"Kari Astala , Martí Prats , Eero Saksman","doi":"10.1016/j.matpur.2024.04.008","DOIUrl":"https://doi.org/10.1016/j.matpur.2024.04.008","url":null,"abstract":"<div><p>We study quasiconformal mappings in planar domains Ω and their regularity properties described in terms of Sobolev, Bessel potential or Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the geometry of the boundary ∂Ω and of the smoothness of the Beltrami coefficient, that guarantee the global regularity of the mappings in these classes. In the Triebel-Lizorkin class with smoothness below 1, the same conditions give global regularity in Ω for the principal solutions with Beltrami coefficient supported in Ω.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000412/pdfft?md5=f62675a2e198d50e0c28eb49218cd39b&pid=1-s2.0-S0021782424000412-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141073033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-06DOI: 10.1016/j.matpur.2024.04.007
David Damanik , Yong Li , Fei Xu
In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the -generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying the higher dimensional discrete convolution operation for several functions: In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for [11], and the combinatorics become harder for larger values of . For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, . Next, we prove existence and uniqueness in the general case , which then covers the remaining cases . As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general .
{"title":"Local existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equation","authors":"David Damanik , Yong Li , Fei Xu","doi":"10.1016/j.matpur.2024.04.007","DOIUrl":"https://doi.org/10.1016/j.matpur.2024.04.007","url":null,"abstract":"<div><p>In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the <span><math><mi>p</mi></math></span>-generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying <strong>the higher dimensional discrete convolution operation for several functions</strong>:<span><span><span><math><munder><munder><mrow><mi>c</mi><mo>×</mo><mo>⋯</mo><mo>×</mo><mi>c</mi></mrow><mo>︸</mo></munder><mrow><mi>p</mi></mrow></munder><mspace></mspace><mo>(</mo><mtext>total distance</mtext><mo>)</mo><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mo>♣</mo></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mo>♣</mo></mrow><mrow><mi>p</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>ν</mi></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mo>♣</mo></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mo>♣</mo></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mspace></mspace><mtext>total distance</mtext></mtd></mtr></mtable></mrow></munder><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>p</mi></mrow></munderover><mi>c</mi><mo>(</mo><msub><mrow><mo>♣</mo></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>.</mo></math></span></span></span> In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> <span>[11]</span>, and the combinatorics become harder for larger values of <span><math><mi>p</mi></math></span>. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, <span><math><mi>p</mi><mo>=</mo><mn>3</mn></math></span>. Next, we prove existence and uniqueness in the general case <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>, which then covers the remaining cases <span><math><mi>p</mi><mo>≥</mo><mn>4</mn></math></span>. As a byproduct, we recover the local result from <span>[11]</span>. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index <em>σ</em> (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general <span><math><mi>p</mi></math></span>.</p>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141164082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-03DOI: 10.1016/j.matpur.2024.04.001
Krzysztof Barański , Yonatan Gutman , Adam Śpiewak
In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k, then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ-almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T-invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space.
在预测混沌系统行为方面，Schroer、Sauer、Ott 和 Yorke 于 1998 年提出了这样的猜想：如果由黎曼流形 X 的光滑差分变换 T 定义的动力系统具有一个吸引子，其信息维度小于 k 的自然度量 μ，那么对一维观测值 h 的 k 次延时测量一般足以对 h 的未来测量进行 μ 几乎确定的预测。在前一篇论文中，我们在欧几里得空间紧凑集 X 的注入式 Lipschitz 变换 T 与遍历 T 不变的 Borel 概率度量 μ 的条件下建立了这一猜想。在本文中，我们证明了具有任意 Borel 概率度量的紧凑集上所有（也是非可逆的）Lipschitz 系统的猜想，并建立了预测不准确的点集度量的衰减率上限。这部分证实了 Schroer、Sauer、Ott 和 Yorke 提出的第二个猜想，该猜想与经验预测算法以及估算被观测系统的维度和所需延迟测量次数（即所谓的嵌入维度）的算法有关。我们还证明了欧几里得空间中伯乐集上局部利普希兹或荷尔德系统的一般时延预测定理。
{"title":"Prediction of dynamical systems from time-delayed measurements with self-intersections","authors":"Krzysztof Barański , Yonatan Gutman , Adam Śpiewak","doi":"10.1016/j.matpur.2024.04.001","DOIUrl":"https://doi.org/10.1016/j.matpur.2024.04.001","url":null,"abstract":"<div><p>In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism <em>T</em> of a Riemannian manifold <em>X</em> admits an attractor with a natural measure <em>μ</em> of information dimension smaller than <em>k</em>, then <em>k</em> time-delayed measurements of a one-dimensional observable <em>h</em> are generically sufficient for <em>μ</em>-almost sure prediction of future measurements of <em>h</em>. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations <em>T</em> of a compact set <em>X</em> in Euclidean space with an ergodic <em>T</em>-invariant Borel probability measure <em>μ</em>. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000345/pdfft?md5=cc88d753949c203cef60a29f0c445bcc&pid=1-s2.0-S0021782424000345-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140948184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.
{"title":"Local solvability and dilation-critical singularities of supercritical fractional heat equations","authors":"Yohei Fujishima , Kotaro Hisa , Kazuhiro Ishige , Robert Laister","doi":"10.1016/j.matpur.2024.04.005","DOIUrl":"https://doi.org/10.1016/j.matpur.2024.04.005","url":null,"abstract":"<div><p>We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a <em>dilation-critical singularity</em> (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140948280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-03DOI: 10.1016/j.matpur.2024.04.006
Alberto Boscaggin , Walter Dambrosio , Guglielmo Feltrin
We consider a perturbation of a central force problem of the form where is a small parameter, and are smooth functions, and U is τ-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem () and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ-periodic solutions bifurcating from invariant tori at . We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential for ). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction.
{"title":"Periodic perturbations of central force problems and an application to a restricted 3-body problem","authors":"Alberto Boscaggin , Walter Dambrosio , Guglielmo Feltrin","doi":"10.1016/j.matpur.2024.04.006","DOIUrl":"10.1016/j.matpur.2024.04.006","url":null,"abstract":"<div><p>We consider a perturbation of a central force problem of the form<span><span><span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>=</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mfrac><mo>+</mo><mi>ε</mi><mspace></mspace><msub><mrow><mi>∇</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>U</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><mi>ε</mi><mo>∈</mo><mi>R</mi></math></span> is a small parameter, <span><math><mi>V</mi><mo>:</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo><mo>→</mo><mi>R</mi></math></span> and <span><math><mi>U</mi><mo>:</mo><mi>R</mi><mo>×</mo><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo><mo>)</mo><mo>→</mo><mi>R</mi></math></span> are smooth functions, and <em>U</em> is <em>τ</em>-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem (<span><math><mi>ε</mi><mo>=</mo><mn>0</mn></math></span>) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular <em>τ</em>-periodic solutions bifurcating from invariant tori at <span><math><mi>ε</mi><mo>=</mo><mn>0</mn></math></span>. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential <span><math><mi>V</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mi>κ</mi><mo>/</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span> for <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>2</mn><mo>)</mo><mo>∖</mo><mo>{</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000394/pdfft?md5=a94afa454e50950cfd681c23244b1192&pid=1-s2.0-S0021782424000394-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-03DOI: 10.1016/j.matpur.2024.04.003
Krzysztof Bogdan , Konstantin Merz
Motivated by the study of relativistic atoms, we consider the Hardy operator acting on functions of the form in , when and . We give a ground state representation of the corresponding form on the half-line (Theorem 1.5). For the proof we use subordinated Bessel heat kernels.
{"title":"Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels","authors":"Krzysztof Bogdan , Konstantin Merz","doi":"10.1016/j.matpur.2024.04.003","DOIUrl":"10.1016/j.matpur.2024.04.003","url":null,"abstract":"<div><p>Motivated by the study of relativistic atoms, we consider the Hardy operator <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mi>κ</mi><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup></math></span> acting on functions of the form <span><math><mi>u</mi><mo>(</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>ℓ</mi></mrow></msup><msub><mrow><mi>Y</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>/</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo></math></span> in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>, when <span><math><mi>κ</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo><mo>∩</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>+</mo><mn>2</mn><mi>ℓ</mi><mo>)</mo></math></span>. We give a ground state representation of the corresponding form on the half-line (<span>Theorem 1.5</span>). For the proof we use subordinated Bessel heat kernels.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000369/pdfft?md5=cef2b60ae87f0ca0fab3ae618e451f4f&pid=1-s2.0-S0021782424000369-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions of the equation where in and . We prove existence of radial solutions which are continuous on in the case , existence of unbounded solutions in the case and a non existence result for . We also give, in the case of Pucci's operators, the explicit value of , which generalizes the Hardy–Sobolev constant for the Laplacian.
本文致力于证明在奇异势存在的情况下，在点球中提出的全非线性均匀椭圆方程的主特征值和相关特征函数的存在性。更确切地说，我们分析了方程F(D2uγ)+λ¯γuγrγ=0 inB(0,1)∖{0},uγ=0 on∂B(0,1) 的解（λ¯γ,uγ）的存在性、唯一性和正则性，其中 uγ>0 in B(0,1)∖{0} 和 γ>0。我们证明了γ<2情况下在B(0,1)‾上连续的径向解的存在性，γ=2情况下无约束解的存在性，以及γ>2情况下的不存在性结果。 我们还给出了普奇算子情况下λ¯2的显式值，它概括了拉普拉斯常数的哈代-索博列夫常数。
{"title":"Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls","authors":"Isabeau Birindelli , Françoise Demengel , Fabiana Leoni","doi":"10.1016/j.matpur.2024.04.004","DOIUrl":"https://doi.org/10.1016/j.matpur.2024.04.004","url":null,"abstract":"<div><p>This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions <span><math><mo>(</mo><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>γ</mi></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>)</mo></math></span> of the equation<span><span><span><math><mi>F</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>)</mo><mo>+</mo><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>γ</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub></mrow><mrow><msup><mrow><mi>r</mi></mrow><mrow><mi>γ</mi></mrow></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mspace></mspace><mrow><mi>in</mi></mrow><mspace></mspace><mi>B</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo><mo>,</mo><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>=</mo><mn>0</mn><mspace></mspace><mrow><mi>on</mi></mrow><mspace></mspace><mo>∂</mo><mi>B</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span></span></span> where <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> in <span><math><mi>B</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> and <span><math><mi>γ</mi><mo>></mo><mn>0</mn></math></span>. We prove existence of radial solutions which are continuous on <span><math><mover><mrow><mi>B</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>‾</mo></mover></math></span> in the case <span><math><mi>γ</mi><mo><</mo><mn>2</mn></math></span>, existence of unbounded solutions in the case <span><math><mi>γ</mi><mo>=</mo><mn>2</mn></math></span> and a non existence result for <span><math><mi>γ</mi><mo>></mo><mn>2</mn></math></span>. We also give, in the case of Pucci's operators, the explicit value of <span><math><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub></math></span>, which generalizes the Hardy–Sobolev constant for the Laplacian.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140947588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-03DOI: 10.1016/j.matpur.2024.04.002
Daniel Lenz , Simon Puchert , Marcel Schmidt
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool is a connection between polar sets in such boundaries and null sets of paths. This connection relies on suitably diverging functions of finite energy.
{"title":"Recurrent and (strongly) resolvable graphs","authors":"Daniel Lenz , Simon Puchert , Marcel Schmidt","doi":"10.1016/j.matpur.2024.04.002","DOIUrl":"https://doi.org/10.1016/j.matpur.2024.04.002","url":null,"abstract":"<div><p>We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool is a connection between polar sets in such boundaries and null sets of paths. This connection relies on suitably diverging functions of finite energy.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000357/pdfft?md5=1428d47f064fd093764bfc8b1a049da0&pid=1-s2.0-S0021782424000357-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140948279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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