{"title":"Stability and instability of solitary-wave solutions for the nonlinear Klein-Gordon equation","authors":"Jing Li , Yue Liu , Yifei Wu , Haohao Zheng","doi":"10.1016/j.jfa.2025.110981","DOIUrl":"10.1016/j.jfa.2025.110981","url":null,"abstract":"<div><div>The nonlinear Klein-Gordon (KG) equation,<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>u</mi><mo>=</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi><mo>,</mo><mspace></mspace><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span></span></span> is shown in the present paper to possess the solitary-wave solutions in the form of <span><math><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>ω</mi><mi>t</mi></mrow></msup><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>ω</mi><mo>,</mo><mover><mrow><mi>c</mi></mrow><mrow><mo>→</mo></mrow></mover></mrow></msub><mo>(</mo><mi>x</mi><mo>−</mo><mover><mrow><mi>c</mi></mrow><mrow><mo>→</mo></mrow></mover><mi>t</mi><mo>)</mo></math></span> with the parameters <em>ω</em> and <span><math><mover><mrow><mi>c</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> satisfying <span><math><mo>|</mo><mi>ω</mi><mo>|</mo><mo><</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mo>|</mo><mover><mrow><mi>c</mi></mrow><mrow><mo>→</mo></mrow></mover><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></math></span> and <span><math><mo>|</mo><mover><mrow><mi>c</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>|</mo><mo><</mo><mn>1</mn></math></span>. By employing a new localized virial identity combined with the coercivity and modulation argument, it is demonstrated here that there exists a critical frequency <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mo>|</mo><mover><mrow><mi>c</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>|</mo><mo>)</mo></math></span> such that these localized solitary waves, when considered as solutions of the initial-value problem for the nonlinear KG equation, is dynamically stable when <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>d</mi></mrow></mfrac><mo>,</mo><mo>|</mo><mi>ω</mi><mo>|</mo><mo>></mo><msub><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mo>|</mo><mover><mrow><mi>c</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>|</mo><mo>)</mo></math></span> and is dynamically unstable when <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>d</mi></mrow></mfrac><mo>,</mo><mn>0</mn><mo><</mo><mo>|</mo><mi>ω</mi><mo>|</mo><mo>≤</mo><msub><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mo>|</mo><mover><mrow><mi>c</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>|</mo><mo>)</mo></math></span> or <span><math><mn>1</mn><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>d</mi></mrow></mfrac><mo>≤</mo><mi>p</mi><mo><</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>4</m","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 110981"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143792169","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-spectral theory for the Laplacian on forms","authors":"Nelia Charalambous , Zhiqin Lu","doi":"10.1016/j.jfa.2025.110976","DOIUrl":"10.1016/j.jfa.2025.110976","url":null,"abstract":"<div><div>In this article, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-spectrum of the Laplacian on <em>k</em>-forms, and also prove the decomposition of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-spectrum depending on the order of the forms. We then show that the resolvent set of an operator such as the Laplacian on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. We conclude by providing a detailed description of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> spectrum of the Laplacian on <em>k</em>-forms over hyperbolic space.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 110976"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143792168","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":"Amenable actions of compact and discrete quantum groups on von Neumann algebras","authors":"K. De Commer, J. De Ro","doi":"10.1016/j.jfa.2025.110973","DOIUrl":"10.1016/j.jfa.2025.110973","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a compact or a discrete quantum group and <span><math><mi>A</mi><mo>⊆</mo><mi>B</mi></math></span> an inclusion of <em>σ</em>-finite <span><math><mi>G</mi></math></span>-dynamical von Neumann algebras. We prove that the <span><math><mi>G</mi></math></span>-inclusion <span><math><mi>A</mi><mo>⊆</mo><mi>B</mi></math></span> is strongly equivariantly amenable if and only if it is equivariantly amenable, using techniques from the theory of non-commutative <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-spaces. In particular, if <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span> is a <span><math><mi>G</mi></math></span>-dynamical von Neumann algebra with <em>A σ</em>-finite, the action <span><math><mi>α</mi><mo>:</mo><mi>A</mi><mo>↶</mo><mi>G</mi></math></span> is strongly (inner) amenable if and only if the action <span><math><mi>α</mi><mo>:</mo><mi>A</mi><mo>↶</mo><mi>G</mi></math></span> is (inner) amenable. This result can be seen as a dynamical generalization of Tomatsu's result on the amenability/co-amenability duality for discrete/compact quantum groups. We also provide the first explicit examples of amenable discrete quantum groups that act non-amenably on a von Neumann algebra.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 110973"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Function theory on quotient domains related to the polydisc","authors":"Mainak Bhowmik , Poornendu Kumar","doi":"10.1016/j.jfa.2025.110978","DOIUrl":"10.1016/j.jfa.2025.110978","url":null,"abstract":"<div><div>Inner functions are the backbone of holomorphic function theory. This paper studies the inner functions on quotient domains of the open unit polydisc, <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, arising from the group action of finite pseudo-reflection groups. Such quotient domains are known to be biholomorphic to the proper image <span><math><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> under certain polynomial maps <span><math><mi>θ</mi><mo>:</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. The main contributions of this paper are as follows:<ul><li><span>(1)</span><span><div>We show that the closed algebra generated by inner functions on <span><math><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> forms a proper subalgebra of <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span>, the algebra of bounded holomorphic functions on <span><math><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>.</div></span></li><li><span>(2)</span><span><div>The set of all rational inner functions on <span><math><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> is shown to be dense in the norm-unit ball of <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span> with respect to the uniform compact-open topology, thereby proving the Carathéodory approximation result.</div></span></li><li><span>(3)</span><span><div>As an application of the Carathéodory approximation theorem, we approximate holomorphic functions on <span><math><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> that are continuous in the closure of <span><math><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> by convex combinations of rational inner functions in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm, thereby obtaining a version of the Fisher's theorem.</div></span></li><li><span>(4)</span><span><div>Given the two approximation results above, establishing a structure for rational inner functions is essential. We have identified the structure of rational inner functions on <span><math><mi>θ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>.<","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 110978"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143792170","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":"Blow up analysis for Keller-Segel system","authors":"Hua Chen, Jian-Meng Li , Kelei Wang","doi":"10.1016/j.jfa.2025.110968","DOIUrl":"10.1016/j.jfa.2025.110968","url":null,"abstract":"<div><div>In this paper we develop a blow up theory for the parabolic-elliptic Keller-Segel system, which can be viewed as a parabolic counterpart to the Liouville equation. This theory is applied to the study of first time singularities, ancient solutions and entire solutions, leading to a description of the blow-up limit in the first problem, and the large scale structure in the other two problems.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 4","pages":"Article 110968"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143800525","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":"A degree counting formula for Fuchsian ODEs with unitarizable monodromy","authors":"Hsin-Yuan Huang , Chang-Shou Lin","doi":"10.1016/j.jfa.2025.110969","DOIUrl":"10.1016/j.jfa.2025.110969","url":null,"abstract":"<div><div>In this paper, we study the problem of whether the monodromy matrices of second-order Fuchsian ordinary differential equations (ODEs) are unitary. Let <em>k</em> be the number of non-integer differences of the local exponents at the singular points of the ODEs. By employing the Leray-Schauder degree formulas for the corresponding curvature equations, we show that under certain assumptions, the degree does not vanish when <span><math><mi>k</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></math></span>, which implies that the corresponding monodromy matrices are unitary. Among others, we show the form of the degree counting formulas. To the best of our knowledge, this is the first work that assigns the Leray-Schauder degree to Fuchsian ODEs from the perspective of the corresponding curvature equations.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 110969"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786173","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":"HCIZ integral formula as unitarity of a canonical map between reproducing kernel spaces","authors":"Martin Miglioli","doi":"10.1016/j.jfa.2025.110977","DOIUrl":"10.1016/j.jfa.2025.110977","url":null,"abstract":"<div><div>In this article we prove that the Harish-Chandra-Itzykson-Zuber (HCIZ) integral formula is equivalent to the unitarity of a canonical map between fixed point spaces of unitary representations on Segal-Bargmann spaces. As a consequence, we provide two new proofs of the HCIZ integral formula and alternative proofs of related results.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 110977"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786171","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":"Cauchy transforms and Szegő projections in dual Hardy spaces: Inequalities and Möbius invariance","authors":"David E. Barrett , Luke D. Edholm","doi":"10.1016/j.jfa.2025.110980","DOIUrl":"10.1016/j.jfa.2025.110980","url":null,"abstract":"<div><div>Dual pairs of interior and exterior Hardy spaces associated to a simple closed Lipschitz planar curve are considered, leading to a Möbius invariant function bounding the norm of the Cauchy transform <strong><em>C</em></strong> from below. This function is shown to satisfy strong rigidity properties and is closely connected via the Berezin transform to the square of the Kerzman-Stein operator. Explicit example calculations are presented. For ellipses, a new asymptotically sharp lower bound on the norm of <strong><em>C</em></strong> is produced.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 110980"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sampling discretization in Orlicz spaces","authors":"Egor Kosov , Sergey Tikhonov","doi":"10.1016/j.jfa.2025.110971","DOIUrl":"10.1016/j.jfa.2025.110971","url":null,"abstract":"<div><div>We obtain new sampling discretization results in Orlicz norms on finite dimensional spaces. As applications, we study sampling recovery problems, where the error of the recovery process is calculated with respect to different Orlicz norms. In particular, we are interested in the recovery by linear and nonlinear methods in the norms close to <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 7","pages":"Article 110971"},"PeriodicalIF":1.7,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815319","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":"Viscosity solution to complex Hessian equations on compact Hermitian manifolds","authors":"Jingrui Cheng, Yulun Xu","doi":"10.1016/j.jfa.2025.110936","DOIUrl":"10.1016/j.jfa.2025.110936","url":null,"abstract":"<div><div>We prove the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold that satisfy a determinant domination condition. This viscosity solution is shown to be unique when the right hand is strictly monotone increasing in terms of the solution. When the right hand side does not depend on the solution, we reduces it to the strict monotonicity of the solvability constant.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 5","pages":"Article 110936"},"PeriodicalIF":1.7,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738471","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}
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