{"title":"Endpoint estimates of fractional integral operators when α > n","authors":"Liang Huang , Hanli Tang , Caifeng Zhang","doi":"10.1016/j.bulsci.2024.103569","DOIUrl":"10.1016/j.bulsci.2024.103569","url":null,"abstract":"<div><div>In this paper we prove the reverse endpoint estimate of fractional integral operators<span><span><span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></mfrac><msubsup><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msubsup><msub><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></msub><mo>≤</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>α</mi></mrow></mfrac><mo>,</mo><mo>∞</mo></mrow></msup></mrow></msub></math></span></span></span> when <span><math><mi>α</mi><mo>></mo><mi>n</mi></math></span> for nonnegative function <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. And we also show that<span><span><span><math><msub><mrow><mo>‖</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>α</mi></mrow></mfrac><mo>,</mo><mo>∞</mo></mrow></msup></mrow></msub><mo>≤</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msubsup><msub><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></msub><mo>,</mo></math></span></span></span> where <span><math><mi>α</mi><mo>></mo><mi>n</mi></math></span>, <span><math><mn>0</mn><mo>≤</mo><mi>f</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> and the constant <span><math><msubsup><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msubsup></math></span> is sharp. Moreover we establish the limiting weak type behaviors for fractional integral operators when <span><math><mi>α</mi><mo>></mo><mi>n</mi></math></span>. Specifically, there holds<span><span><span><math><munder><mi>lim</mi><mrow><mi>λ</mi><mo>→</mo><mn>0</mn></mrow></munder><mo></mo><mi>λ</mi><mo>|</mo><mo>{</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>:</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo><</mo><mi>λ</mi><mo>}</mo><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mi>n</mi><mo>−</mo><mi>α</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup><mo>=</mo><mo>∞</mo><mspace></mspace><mtext> for any </mtext","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103569"},"PeriodicalIF":1.3,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173500","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}
Kumi Kobata , Tatsushi Tanaka , Noriko Wakabayashi
{"title":"Rooted tree polylogarithms","authors":"Kumi Kobata , Tatsushi Tanaka , Noriko Wakabayashi","doi":"10.1016/j.bulsci.2024.103568","DOIUrl":"10.1016/j.bulsci.2024.103568","url":null,"abstract":"<div><div>We study multiple polylogarithms indexed by rooted forests and show some properties of them from a viewpoint of the Connes-Kreimer Hopf algebra of rooted trees <span><math><mi>H</mi></math></span>. In particular, we show that almost all primitive elements in <span><math><mi>H</mi></math></span> give relations among the polylogarithms. Moreover, we compare the algebraic structure of the polylogarithms with that of rooted tree maps.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103568"},"PeriodicalIF":1.3,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174289","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":"Abel ordinary differential equation, differential algebra and projective connections","authors":"Oumar Wone","doi":"10.1016/j.bulsci.2024.103567","DOIUrl":"10.1016/j.bulsci.2024.103567","url":null,"abstract":"<div><div>We firstly study the Abel ordinary differential equations of the first and second kind from the perspectives of differential algebra. Then using differential geometry we exhibit some relations between the Abel ordinary differential equation of the first kind and projective connections on surfaces which allow us to find a “Darboux first integral” of the Abel differential equation.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103567"},"PeriodicalIF":1.3,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174290","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":"Smooth linearization of nonautonomous dynamics on the line","authors":"Tian Wang , Zhihua Ren , Jiazhong Yang","doi":"10.1016/j.bulsci.2024.103566","DOIUrl":"10.1016/j.bulsci.2024.103566","url":null,"abstract":"<div><div>The aim of this paper is to study linearization for a sequence of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>α</mi></mrow></msup></math></span> <span><math><mo>(</mo><mi>r</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span> maps on the line corresponding to a class of non-autonomous dynamics with discrete time. We obtain the following results: (i) if <span><math><mi>r</mi><mo>+</mo><mi>α</mi><mo>></mo><mn>1</mn></math></span>, then there exists a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>α</mi></mrow></msup></math></span> linearization for <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>α</mi></mrow></msup></math></span> hyperbolic systems; (ii) for <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> hyperbolic systems, then there is a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>β</mi></mrow></msup></math></span> linearization for any <em>β</em> with <span><math><mn>0</mn><mo><</mo><mi>β</mi><mo><</mo><mn>1</mn></math></span>. Moreover, by presenting a concrete example, we demonstrate that in case (ii), the result is the best. As a special case, we also present a detailed investigation on periodic difference equations in this paper.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103566"},"PeriodicalIF":1.3,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173076","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":"Holomorphic factorization of vector bundle automorphisms","authors":"George Ioniţă, Frank Kutzschebauch","doi":"10.1016/j.bulsci.2024.103565","DOIUrl":"10.1016/j.bulsci.2024.103565","url":null,"abstract":"<div><div>We prove that any null-homotopic special holomorphic vector bundle automorphism of a holomorphic rank 2 vector bundle <em>E</em> over a Stein space <em>X</em> can be written as a finite product of unipotent holomorphic vector bundle automorphisms as well as a finite product of exponentials.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103565"},"PeriodicalIF":1.3,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A new approach to approximate the solution of two general functional equations in quasi-Banach spaces","authors":"Jawad Boutarfass, Iz-iddine EL-Fassi, Lahcen Oukhtite","doi":"10.1016/j.bulsci.2024.103564","DOIUrl":"10.1016/j.bulsci.2024.103564","url":null,"abstract":"<div><div>In this paper, we first establish a stability result for a functional equation in single variable in complete <em>b</em>-metric spaces. This result can be applied to prove the stability of various functional equations in quasi-Banach spaces. The perturbation of Schröder equation in quasi-Banach spaces is also proved. As an application of our main result, the stability in the sense of “G.-L. Forti and P. Gǎvruta” for two general functional equations in quasi-Banach spaces is studied. These equations generalize, among others, those characterizing multi-additive and multi-quadratic functions. The present findings extend and generalize the recent main results presented in Ciepliński (2023) <span><span>[12]</span></span> and their corollaries.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103564"},"PeriodicalIF":1.3,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173030","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":"Boundedness of maximal operators and Sobolev inequalities on Musielak-Orlicz spaces over unbounded metric measure spaces","authors":"Takao Ohno , Tetsu Shimomura","doi":"10.1016/j.bulsci.2024.103546","DOIUrl":"10.1016/j.bulsci.2024.103546","url":null,"abstract":"<div><div>We prove the boundedness of the Hardy–Littlewood maximal operator <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>λ</mi><mo>≥</mo><mn>1</mn></math></span>, on Musielak-Orlicz spaces <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>Φ</mi></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> over unbounded metric measure spaces as an extension of earlier results, where <em>λ</em> is its modification rate. As an application of the boundedness of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span>, we establish a generalization of Sobolev inequalities for the variable Riesz potentials <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>,</mo><mi>τ</mi></mrow></msub><mi>f</mi><mo>,</mo><mspace></mspace><mi>τ</mi><mo>≥</mo><mn>1</mn></math></span>, on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>Φ</mi></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> over unbounded metric measure spaces, where <em>τ</em> is its modification rate. As a corollary, we show the boundedness of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> and Sobolev inequalities for <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>,</mo><mi>τ</mi></mrow></msub><mi>f</mi></math></span> for double phase functionals with variable exponents. Our results are new even for the doubling metric measure setting in that the underlying spaces need not be bounded.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103546"},"PeriodicalIF":1.3,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173033","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}
Mohammad Hossein Keshavarz , Yefei Ren , Guodong Zhou
{"title":"Dominant and codominant dimensions for quiver representations","authors":"Mohammad Hossein Keshavarz , Yefei Ren , Guodong Zhou","doi":"10.1016/j.bulsci.2024.103563","DOIUrl":"10.1016/j.bulsci.2024.103563","url":null,"abstract":"<div><div>Let <span><math><mi>M</mi></math></span> be a module category and <span><math><mi>Q</mi></math></span> a rooted quiver. In this paper, we study the dominant (resp. codominant) dimension of the category <span><math><mrow><mi>Rep</mi></mrow><mo>(</mo><mi>Q</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> of <span><math><mi>M</mi></math></span>-valued representations of <span><math><mi>Q</mi></math></span>. To do this, we first study injective envelopes and projective covers that play important roles in homological algebra and give explicit formulas for them in the category <span><math><mrow><mi>Rep</mi></mrow><mo>(</mo><mi>Q</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span>, whose origins go back to the classical representation theory of a finite quiver over a field. Then, by using such descriptions, we compute the dominant (resp. codominant) dimension of <span><math><mrow><mi>Rep</mi></mrow><mo>(</mo><mi>Q</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span>.</div><div>We show that the dominant dimension of <span><math><mrow><mi>Rep</mi></mrow><mo>(</mo><mi>Q</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> is at most one for every nonzero module category <span><math><mi>M</mi></math></span> and any right rooted quiver with at least one arrow.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103563"},"PeriodicalIF":1.3,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173031","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":"Dynamics near the three-point heteroclinic cycles with saddle-focus","authors":"Duo Hua, Xingbo Liu","doi":"10.1016/j.bulsci.2024.103562","DOIUrl":"10.1016/j.bulsci.2024.103562","url":null,"abstract":"<div><div>This paper studies the bifurcation phenomena of heteroclinic cycles connecting three equilibria in a three-dimensional vector field. Based on Lin's method, we prove the existence of shift dynamics near the three-point heteroclinic cycle, showing the existence of chaotic behavior. Moreover, we present more details about the bifurcation results, such as the existence of a three-point heteroclinic cycle, two-point heteroclinic cycles, homoclinic cycles and 1-periodic orbits bifurcated from the primary three-point heteroclinic cycle. Furthermore, the coexistence of 1-periodic orbit and homoclinic cycle, and the coexistence of 1-periodic orbit and two-point heteroclinic cycle are proved respectively.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103562"},"PeriodicalIF":1.3,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173032","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 a pointwise inequality for even Legendre polynomials in high dimensional spheres","authors":"Shirong Chen , Yi C. Huang , Jian-Yang Zhang","doi":"10.1016/j.bulsci.2024.103545","DOIUrl":"10.1016/j.bulsci.2024.103545","url":null,"abstract":"<div><div>We present a pointwise inequality for adjacent even Legendre polynomials in high dimensional spheres featuring the effect of spectral gaps. This improves a recent result of Imbert, Silvestre and Villani that is crucially used in their study of the Fisher information for the Boltzmann equation.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103545"},"PeriodicalIF":1.3,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173035","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}