Journal of Mathematical Analysis and Applications最新文献

{"title":"Restricted slowly growing digits for infinite iterated function systems","authors":"Gerardo González Robert ,&nbsp;Mumtaz Hussain ,&nbsp;Nikita Shulga ,&nbsp;Hiroki Takahasi","doi":"10.1016/j.jmaa.2025.129478","DOIUrl":"10.1016/j.jmaa.2025.129478","url":null,"abstract":"<div><div>For an infinite iterated function system <strong>f</strong> on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> with an attractor <span><math><mi>Λ</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> and for an infinite subset <span><math><mi>D</mi><mo>⊆</mo><mi>N</mi></math></span>, consider the set<span><span><span><math><mi>E</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>D</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>x</mi><mo>∈</mo><mi>Λ</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>D</mi><mspace></mspace><mtext>for all</mtext><mspace></mspace><mi>n</mi><mo>∈</mo><mi>N</mi><mspace></mspace><mtext>and</mtext><mspace></mspace><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo>⁡</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mo>∞</mo><mo>}</mo><mo>.</mo></math></span></span></span> For a function <span><math><mi>φ</mi><mo>:</mo><mi>N</mi><mo>→</mo><mo>[</mo><mi>min</mi><mo>⁡</mo><mi>D</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> such that <span><math><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>→</mo><mo>∞</mo></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>, we compute the Hausdorff dimension of the set<span><span><span><math><mi>S</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>D</mi><mo>,</mo><mi>φ</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>D</mi><mo>)</mo><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>≤</mo><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mspace></mspace><mtext>for all</mtext><mspace></mspace><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></mrow><mo>.</mo></math></span></span></span> We prove that the Hausdorff dimension stays the same no matter how slowly the function <em>φ</em> grows. One of the consequences of our result is the recent work of Takahasi (2023), which only dealt with regular continued fraction expansions. We further extend our result to slowly growing products of (not necessarily consecutive) digits.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 1","pages":"Article 129478"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644705","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":"New type solutions for a biharmonic Hénon problem with slightly subcritical Sobolev exponent","authors":"Wenjing Chen,&nbsp;Fang Yu","doi":"10.1016/j.jmaa.2025.129481","DOIUrl":"10.1016/j.jmaa.2025.129481","url":null,"abstract":"<div><div>In this paper, we study the following biharmonic Hénon problem<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mi>K</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi></mrow></msup><mi>u</mi><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mi>in</mi></mrow><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mi>on</mi></mrow><mspace></mspace><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></span></span></span> where Ω is a bounded and smooth domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mn>6</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>12</mn></math></span>, <span><math><mi>p</mi><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow></mfrac></math></span>, and <span><math><mi>p</mi><mo>+</mo><mn>1</mn><mo>=</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow></mfrac></math></span> denotes the critical Sobolev exponent for the embedding <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>∩</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>↪</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>. The parameter <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span> is a small, and the function <span><math><mi>K</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span> is positive and satisfies<span><span><span><math><mi>∇</mi><mo>(</mo><mi>K</mi><msup><mrow><mo>(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>)</mo></mrow><mrow><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>)</mo><mo>⋅</mo><mi>η</mi><mo>(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>)</mo><mo>&gt;</mo><mn>0</mn><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>∈</mo><mo>∂</mo><mi>Ω</mi></math></span> is a non-degenerate critical point of <em>K</em> which is restricted to the boundary of Ω, and <em>η</em> is the inner normal unit vector on ∂Ω. We establish the existence of a positive solution and a sign-changing solution with two bubbles concentrating at <span><math><msub><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> for the above problem.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 1","pages":"Article 129481"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644706","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":"Dirac structure for linear dynamical systems on Sobolev spaces","authors":"N. Kumar ,&nbsp;H.J. Zwart ,&nbsp;J.J.W. van der Vegt","doi":"10.1016/j.jmaa.2025.129493","DOIUrl":"10.1016/j.jmaa.2025.129493","url":null,"abstract":"<div><div>The port-Hamiltonian structure of linear dynamical systems is defined by a Dirac structure. In this paper we prove existence and well-posedness of a Dirac structure for linear dynamical systems on Sobolev spaces of differential forms on a bounded, connected and oriented manifold with Lipschitz continuous boundary. This result extends the proof of a Dirac structure for linear dynamical systems originally defined on smooth differential forms to a much larger class of function spaces, which is of theoretical importance and provides a solid basis for the numerical discretization of many linear port-Hamiltonian dynamical systems.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 2","pages":"Article 129493"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704571","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":"On the existence of eigenvalues of a one-dimensional Dirac operator","authors":"Daniel Sánchez-Mendoza ,&nbsp;Monika Winklmeier","doi":"10.1016/j.jmaa.2025.129485","DOIUrl":"10.1016/j.jmaa.2025.129485","url":null,"abstract":"<div><div>The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove existence of such eigenvalues, estimate how many eigenvalues there are, and give upper and lower bounds for them.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 2","pages":"Article 129485"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644157","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 Cartesian product of shrinking target sets in dyadic system and triadic system","authors":"Wanjin Cheng","doi":"10.1016/j.jmaa.2025.129495","DOIUrl":"10.1016/j.jmaa.2025.129495","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we consider the Cartesian product of shrinking target sets. Let &lt;em&gt;f&lt;/em&gt; and &lt;em&gt;g&lt;/em&gt; be two positive continuous functions. For any &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, we define the shrinking target sets as follows:&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mtext&gt; for infinitely many &lt;/mtext&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mtext&gt; for infinitely many &lt;/mtext&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denote the Birkhoff ergodic sums, and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtext&gt;mod &lt;/mtext&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;The Hausdorff dimension of the Cartesian product set &lt;span&gt;&lt;math","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 1","pages":"Article 129495"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683950","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":"Polynomial growth and functional calculus in algebras of integrable cross-sections","authors":"Felipe I. Flores","doi":"10.1016/j.jmaa.2025.129486","DOIUrl":"10.1016/j.jmaa.2025.129486","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a locally compact group with polynomial growth of order <em>d</em>, a polynomial weight <em>ν</em> on <span><math><mi>G</mi></math></span> and a Fell bundle <span><math><mi>C</mi><mover><mo>→</mo><mi>q</mi></mover><mi>G</mi></math></span>. We study the Banach ^⁎-algebras <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>G</mi><mspace></mspace><mo>|</mo><mspace></mspace><mi>C</mi><mo>)</mo></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>ν</mi></mrow></msup><mo>(</mo><mi>G</mi><mspace></mspace><mo>|</mo><mspace></mspace><mi>C</mi><mo>)</mo></math></span>, consisting of integrable cross-sections with respect to <span><math><mi>d</mi><mi>x</mi></math></span> and <span><math><mi>ν</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi></math></span>, respectively. By exploring new relations between the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-norms and the norm of the Hilbert <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-module <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>G</mi><mspace></mspace><mo>|</mo><mspace></mspace><mi>C</mi><mo>)</mo></math></span>, we are able to show that the growth of the self-adjoint, compactly supported, continuous cross-sections is polynomial. More precisely, they satisfy<span><span><span><math><mo>‖</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>t</mi><mi>Φ</mi></mrow></msup><mo>‖</mo><mo>=</mo><mi>O</mi><mo>(</mo><mo>|</mo><mi>t</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mtext>as</mtext><mspace></mspace><mo>|</mo><mi>t</mi><mo>|</mo><mo>→</mo><mo>∞</mo><mo>,</mo></math></span></span></span> for values of <em>n</em> that only depend on <em>d</em> and the weight <em>ν</em>. We use this fact to develop a smooth functional calculus for such elements. We also give some sufficient conditions for these algebras to be symmetric. As consequences, we show that these algebras are locally regular, ^⁎-regular and have the Wiener property (when symmetric), among other results. Our results are already new for convolution algebras associated with <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-dynamical systems.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 2","pages":"Article 129486"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644543","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":"Limits of hypercyclic operators on Hilbert spaces","authors":"Pietro Aiena,&nbsp;Fabio Burderi,&nbsp;Salvatore Triolo","doi":"10.1016/j.jmaa.2025.129484","DOIUrl":"10.1016/j.jmaa.2025.129484","url":null,"abstract":"<div><div>This article concerns the operators <span><math><mi>T</mi><mo>∈</mo><mi>L</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, defined on a separable Hilbert space <em>H</em>, that belong to the norm closure <span><math><mover><mrow><mi>H</mi><mi>C</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>‾</mo></mover></math></span> in <span><math><mi>L</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> of the set <span><math><mi>H</mi><mi>C</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> of all hypercyclic operators. Starting from a Herrero's characterization of these operators <span><span>[11]</span></span> we deduce some criteria that are very useful in many concrete cases. We also show that if <span><math><mi>T</mi><mo>∈</mo><mi>L</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is invertible then <span><math><mi>T</mi><mo>∈</mo><mover><mrow><mi>H</mi><mi>C</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>‾</mo></mover></math></span> if and only if <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><mover><mrow><mi>H</mi><mi>C</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>‾</mo></mover></math></span>. This result extends to <span><math><mover><mrow><mi>H</mi><mi>C</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>‾</mo></mover></math></span> a known result of Kitai and Herrero established for hypercyclic operators, (<span><span>[13]</span></span>).</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129484"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737921","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":"An alternate proof for the global mean speed of bistable transition fronts","authors":"Linlin Li ,&nbsp;Hong Xu ,&nbsp;Zhi-Cheng Wang","doi":"10.1016/j.jmaa.2025.129492","DOIUrl":"10.1016/j.jmaa.2025.129492","url":null,"abstract":"<div><div>In this paper, we present an alternate proof to show the existence and uniqueness of the global mean speed of bistable transition fronts under a general framework.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 2","pages":"Article 129492"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644593","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":"(p,q)-Sobolev inequality and Nash inequality on compact Finsler metric measure manifolds","authors":"Xinyue Cheng,&nbsp;Qihui Ni","doi":"10.1016/j.jmaa.2025.129491","DOIUrl":"10.1016/j.jmaa.2025.129491","url":null,"abstract":"<div><div>In this paper, we carry out in-depth research centering around the <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-Sobolev inequality and Nash inequality on compact Finsler metric measure manifolds under the condition that <span><math><msub><mrow><mi>Ric</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>≥</mo><mo>−</mo><mi>K</mi></math></span> for some <span><math><mi>K</mi><mo>≥</mo><mn>0</mn></math></span>. We first obtain a global <em>p</em>-Poincaré inequality on complete Finsler manifolds. Based on this, we can derive a <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-Sobolev inequality. Furthermore, we establish a global optimal <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-Sobolev inequality. Finally, as an application of the <em>p</em>-Poincaré inequality, we prove a Nash inequality.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 2","pages":"Article 129491"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644594","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":"Upper bounds for the blow-up time of the 2-d parabolic-elliptic Patlak-Keller-Segel model of chemotaxis","authors":"Patrick Maheux","doi":"10.1016/j.jmaa.2025.129487","DOIUrl":"10.1016/j.jmaa.2025.129487","url":null,"abstract":"<div><div>In this paper, we obtain upper bounds for the critical time <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> of the blow-up for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. No moment condition or/and entropy condition are required on the initial data; only the usual assumptions of non-negativity and finiteness of the total mass is assumed. The result is expressed not only in terms of supercritical mass <span><math><mi>M</mi><mo>&gt;</mo><mn>8</mn><mi>π</mi></math></span>, but also in terms of the <em>shape</em> of the initial data.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 2","pages":"Article 129487"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644463","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}