Journal of Functional Analysis最新文献

{"title":"Cartan semigroups and twisted groupoid C*-algebras","authors":"Tristan Bice ,&nbsp;Lisa Orloff Clark ,&nbsp;Ying-Fen Lin ,&nbsp;Kathryn McCormick","doi":"10.1016/j.jfa.2025.111038","DOIUrl":"10.1016/j.jfa.2025.111038","url":null,"abstract":"<div><div>We prove that twisted groupoid C*-algebras are characterised, up to isomorphism, by having <em>Cartan semigroups</em>, a natural generalisation of normaliser semigroups of Cartan subalgebras. This extends the classic Kumjian-Renault theory to general twisted étale groupoid C*-algebras, even non-reduced C*-algebras of non-effective groupoids.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 7","pages":"Article 111038"},"PeriodicalIF":1.7,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity theory and geometry of unbalanced optimal transport","authors":"Thomas Gallouët ,&nbsp;Roberta Ghezzi ,&nbsp;François-Xavier Vialard","doi":"10.1016/j.jfa.2025.111042","DOIUrl":"10.1016/j.jfa.2025.111042","url":null,"abstract":"<div><div>Using the dual formulation only, we show that the regularity of unbalanced optimal transport, also called entropy-transport, inherits from the regularity of standard optimal transport. We provide detailed examples of Riemannian manifolds and costs for which unbalanced optimal transport is regular. Among all entropy-transport formulations, the Wasserstein-Fisher-Rao (WFR) metric, also called Hellinger-Kantorovich, stands out since it admits a dynamical formulation, which extends the Benamou-Brenier formulation of optimal transport. After demonstrating the equivalence between dynamical and static formulations on a closed Riemannian manifold, we prove a polar factorization theorem, similar to the one due to Brenier and Mc-Cann. As a byproduct, we formulate the Monge-Ampère equation associated with the WFR metric, which also holds for more general costs. Last, we study the link between <em>c</em>-convex functions for the cost induced by the WFR metric and the cost on the cone. The main result is that the weak Ma-Trudinger-Wang condition on the cone implies the same condition on the manifold for the cost induced by the WFR metric.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 7","pages":"Article 111042"},"PeriodicalIF":1.7,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space","authors":"Yaoting Gui ,&nbsp;Yuqiao Li ,&nbsp;Jun Sun","doi":"10.1016/j.jfa.2025.111033","DOIUrl":"10.1016/j.jfa.2025.111033","url":null,"abstract":"<div><div>We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is sufficiently small. Then we show that from sufficiently large initial coordinate sphere, the area preserving mean curvature flow exists for all time and converges exponentially fast to a constant mean curvature surface in 3-dimensional asymptotically Schwarzschild spaces. This provides a new approach to the existence of foliation established by Huisken and Yau (<span><span>[11]</span></span>). And also a uniqueness result follows.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 7","pages":"Article 111033"},"PeriodicalIF":1.7,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Wave map null form estimates via Peter–Weyl theory","authors":"Grigalius Taujanskas","doi":"10.1016/j.jfa.2025.111040","DOIUrl":"10.1016/j.jfa.2025.111040","url":null,"abstract":"<div><div>We study spacetime estimates for the wave map null form <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> on <span><math><mi>R</mi><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. By using the Lie group structure of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and Peter–Weyl theory, combined with the time-periodicity of the conformal wave equation on <span><math><mi>R</mi><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, we extend the classical ideas of Klainerman and Machedon to estimates on <span><math><mi>R</mi><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, allowing for a range of powers of natural (Laplacian and wave) Fourier multiplier operators. A key difference in these curved space estimates as compared to the flat case is a loss of an arbitrarily small amount of differentiability, attributable to a lack of dispersion of linear waves on <span><math><mi>R</mi><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. This arises in Fourier space from the product structure of irreducible representations of <span><math><mrow><mi>SU</mi></mrow><mo>(</mo><mn>2</mn><mo>)</mo></math></span>. We further show that our estimates imply weighted estimates for the null form on Minkowski space.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 111040"},"PeriodicalIF":1.7,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On positivity of the Q-curvatures of conformal metrics","authors":"Mingxiang Li ,&nbsp;Xingwang Xu","doi":"10.1016/j.jfa.2025.111011","DOIUrl":"10.1016/j.jfa.2025.111011","url":null,"abstract":"<div><div>We mainly show that for a conformal metric <span><math><mi>g</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></mfrac></mrow></msup><mo>|</mo><mi>d</mi><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></math></span>, if the <span><math><mn>2</mn><mi>m</mi><mo>−</mo></math></span>order Q-curvature <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>)</mo></mrow></msubsup></math></span> is positive and has slow decay barrier near infinity, the lower order Q-curvature <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></msubsup></math></span> are both positive if <em>m</em> is at least two.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 8","pages":"Article 111011"},"PeriodicalIF":1.7,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: An entropy formulation","authors":"Kai Du ,&nbsp;Ruoyang Liu","doi":"10.1016/j.jfa.2025.111012","DOIUrl":"10.1016/j.jfa.2025.111012","url":null,"abstract":"<div><div>In this paper, we establish the existence, uniqueness and stability results for the obstacle problem associated with a degenerate nonlinear diffusion equation perturbed by conservative gradient noise. Our approach revolves round introducing a new entropy formulation for stochastic variational inequalities. As a consequence, we obtain a novel well-posedness result for the obstacle problem of deterministic porous medium equations with nonlinear reaction terms.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 8","pages":"Article 111012"},"PeriodicalIF":1.7,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lp-boundedness of wave operators for fourth order Schrödinger operators with zero resonances on R3","authors":"Haruya Mizutani ,&nbsp;Zijun Wan ,&nbsp;Xiaohua Yao","doi":"10.1016/j.jfa.2025.111013","DOIUrl":"10.1016/j.jfa.2025.111013","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be the fourth-order Schrödinger operator on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with a real-valued fast-decaying potential &lt;em&gt;V&lt;/em&gt;. If zero is neither a resonance nor an eigenvalue of &lt;em&gt;H&lt;/em&gt;, then it was recently shown that the wave operators &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are bounded on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and unbounded at the endpoints &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;This paper is to further establish the &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-boundedness of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; that exhibit all types of singularities at the zero energy threshold. We first prove that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are bounded on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; in the first kind resonance case, and then proceed to establish for the second kind resonance case that they are bounded on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, but not if &lt;span&gt;&lt;math&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In the third kind resonance case, we also show that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are bounded on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and generically unbounded on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 8","pages":"Article 111013"},"PeriodicalIF":1.7,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a Hardy–Morrey inequality","authors":"Ryan Hynd ,&nbsp;Simon Larson ,&nbsp;Erik Lindgren","doi":"10.1016/j.jfa.2025.111002","DOIUrl":"10.1016/j.jfa.2025.111002","url":null,"abstract":"<div><div>Morrey's classical inequality implies the Hölder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality<span><span><span><math><mi>λ</mi><msubsup><mrow><mo>‖</mo><mfrac><mrow><mi>u</mi></mrow><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>n</mi><mo>/</mo><mi>p</mi></mrow></msubsup></mrow></mfrac><mo>‖</mo></mrow><mrow><mo>∞</mo></mrow><mrow><mi>p</mi></mrow></msubsup><mo>≤</mo><munder><mo>∫</mo><mrow><mi>Ω</mi></mrow></munder><mo>|</mo><mi>D</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi></math></span></span></span> for any open set <span><math><mi>Ω</mi><mo>⊊</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. This inequality is valid for functions supported in Ω and with <em>λ</em> a positive constant independent of <em>u</em>. The crucial hypothesis is that the exponent <em>p</em> exceeds the dimension <em>n</em>. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of Ω, sharp constants, and the existence of a nontrivial <em>u</em> which saturates the inequality.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 111002"},"PeriodicalIF":1.7,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143881719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The direct moving sphere for fractional Laplace equation","authors":"Congming Li ,&nbsp;Meiqing Xu ,&nbsp;Hui Yang ,&nbsp;Ran Zhuo","doi":"10.1016/j.jfa.2025.111010","DOIUrl":"10.1016/j.jfa.2025.111010","url":null,"abstract":"<div><div>This paper works on the direct method of moving spheres and establishes a Liouville-type theorem for the fractional elliptic equation<span><span><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><mi>u</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mtext>in </mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span></span></span> with general non-linearity. One of the key improvements over the previous work is that we do not require the usual Lipschitz condition. In fact, we only assume the structural condition that <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>α</mi></mrow></mfrac></mrow></msup></math></span> is monotonically decreasing. This differs from the usual approach such as Chen-Li-Li (Adv. Math. 2017), which needs the Lipschitz condition on <em>f</em>, or Chen-Li-Zhang (J. Funct. Anal. 2017), which relies on both the structural condition and the monotonicity of <em>f</em>. We also use the direct moving spheres method to give an alternative proof for the Liouville-type theorem of the fractional Lane-Emden equation in a half space. Similarly, our proof does not depend on the integral representation of solutions compared to existing ones. The methods developed here should also apply to problems involving more general non-local operators, especially if no equivalent integral equations exist.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 8","pages":"Article 111010"},"PeriodicalIF":1.7,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonlocal Liouville theorems with gradient nonlinearity","authors":"Anup Biswas ,&nbsp;Alexander Quaas ,&nbsp;Erwin Topp","doi":"10.1016/j.jfa.2025.111008","DOIUrl":"10.1016/j.jfa.2025.111008","url":null,"abstract":"<div><div>In this article we consider a large family of nonlinear nonlocal equations involving gradient nonlinearity and provide a unified approach, based on the Ishii-Lions type technique, to establish Liouville properties of the solutions. We also answer an open problem raised by Cirant and Goffi <span><span>[24]</span></span>. Some applications to regularity issues are also studied.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 8","pages":"Article 111008"},"PeriodicalIF":1.7,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}