Journal of Functional Analysis最新文献

{"title":"Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow","authors":"Jitao Xu,&nbsp;Lifeng Zhao","doi":"10.1016/j.jfa.2024.110704","DOIUrl":"10.1016/j.jfa.2024.110704","url":null,"abstract":"<div><div>In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> to <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with arbitrary given coefficients <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><mi>R</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&gt;</mo><mn>0</mn></math></span>. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schrödinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functional. However, the blowup rate is independent of the coefficients.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110704"},"PeriodicalIF":1.7,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142528660","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 the classification of function algebras on subvarieties of noncommutative operator balls","authors":"Jeet Sampat,&nbsp;Orr Moshe Shalit","doi":"10.1016/j.jfa.2024.110703","DOIUrl":"10.1016/j.jfa.2024.110703","url":null,"abstract":"<div><div>We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110703"},"PeriodicalIF":1.7,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142528659","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":"Continuous asymmetric Doob inequalities in noncommutative symmetric spaces","authors":"Yong Jiao,&nbsp;Hui Li,&nbsp;Sijie Luo,&nbsp;Lian Wu","doi":"10.1016/j.jfa.2024.110701","DOIUrl":"10.1016/j.jfa.2024.110701","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be a noncommutative probability space equipped with a filtration &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&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;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; whose union is &lt;span&gt;&lt;math&gt;&lt;msup&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;/msup&gt;&lt;/math&gt;&lt;/span&gt;-dense in &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&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;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the associated conditional expectations. We prove in the present paper that if the symmetric space &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Int&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&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;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &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;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;em&gt;E&lt;/em&gt; is &lt;span&gt;&lt;math&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-convex and &lt;em&gt;w&lt;/em&gt;-concave with &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, then the following holds:&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&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;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; provided &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. Similar result holds for &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, if &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Int&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&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;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &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;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;em&gt;E&lt;/em&gt; is &lt;em&gt;w&lt;/em&gt;-concave with &lt;span&gt;&lt;math&gt;&lt;mn&gt;2&lt;/","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110701"},"PeriodicalIF":1.7,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445751","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":"Functional L1-Lp inequalities in the CAR algebra","authors":"Yong Jiao,&nbsp;Sijie Luo,&nbsp;Dejian Zhou","doi":"10.1016/j.jfa.2024.110700","DOIUrl":"10.1016/j.jfa.2024.110700","url":null,"abstract":"<div><div>In the present paper, we use the semigroup method to investigate various functional inequalities invoking <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> norms in the framework of canonical anti-commuting relations algebra (CAR algebra for short). As the main results, we obtain the Poincaré inequality for Talagrand type sum, Eldan-Gross inequality for projections, and the Talagrand influence inequality along with its strengthening form in the CAR algebra. All our results strengthen the noncommutative Poincaré inequality of Efraim and Lust-Piquard at several points. We conclude the paper with two applications of our inequalities. In the first application, we apply the noncommutative Eldan-Gross inequality to derive two KKL-type inequalities in the CAR algebra, which are closely related to the quantum KKL conjecture of Montanaro and Osborne. The second application is the CAR algebra counterpart of the superconcentration phenomenon derived from the noncommutative Talagrand influence inequality.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110700"},"PeriodicalIF":1.7,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142528656","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":"Inclusions of simple C⁎-algebras arising from compact group actions","authors":"Miho Mukohara","doi":"10.1016/j.jfa.2024.110702","DOIUrl":"10.1016/j.jfa.2024.110702","url":null,"abstract":"<div><div>Inclusions of operator algebras have long been studied. In particular, inclusions arising from actions of compact groups on factors were studied by Izumi-Longo-Popa and others. The correspondence between intermediate subfactors and subgroups is called the Galois correspondence. Analogues for actions on C^⁎-algebras have been studied by Izumi, Cameron-Smith, Peligrad, and others. In this article, we give examples of compact group actions on simple C^⁎-algebras for which the Galois correspondence holds.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110702"},"PeriodicalIF":1.7,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142528658","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 dual Kadec norms","authors":"Petr Hájek","doi":"10.1016/j.jfa.2024.110698","DOIUrl":"10.1016/j.jfa.2024.110698","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mo>‖</mo><mo>⋅</mo><mo>‖</mo><mo>)</mo></math></span> be a Banach space such that all <span><math><msup><mrow><mi>w</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-convergent sequences in the dual unit sphere <span><math><msub><mrow><mi>S</mi></mrow><mrow><msup><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mrow></msub></math></span> are also norm convergent. Then the weak^⁎ and norm topologies agree on <span><math><msub><mrow><mi>S</mi></mrow><mrow><msup><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mrow></msub></math></span>. By known results it implies that <em>X</em> has a renorming whose dual is locally uniformly rotund, hence also <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-Fréchet smooth. In particular, <em>X</em> is an Asplund space. Our results also lend an alternative proof of the celebrated Josefson-Nissenzweig theorem.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110698"},"PeriodicalIF":1.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433478","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":"Imprimitivity theorems and self-similar actions on Fell bundles","authors":"Anna Duwenig ,&nbsp;Boyu Li","doi":"10.1016/j.jfa.2024.110699","DOIUrl":"10.1016/j.jfa.2024.110699","url":null,"abstract":"<div><div>We introduce the notion of self-similar actions of groupoids on other groupoids and Fell bundles. This leads to a new imprimitivity theorem arising from such dynamics, generalizing many earlier imprimitivity theorems involving group and groupoid actions.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110699"},"PeriodicalIF":1.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445750","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":"Quantitative observability for one-dimensional Schrödinger equations with potentials","authors":"Pei Su ,&nbsp;Chenmin Sun ,&nbsp;Xu Yuan","doi":"10.1016/j.jfa.2024.110695","DOIUrl":"10.1016/j.jfa.2024.110695","url":null,"abstract":"<div><div>In this note, we prove the quantitative observability with an explicit control cost for the 1D Schrödinger equation over <span><math><mi>R</mi></math></span> with real-valued, bounded continuous potential on thick sets. Our proof relies on different techniques for low-frequency and high-frequency estimates. In particular, we extend the large time observability result for the 1D free Schrödinger equation in Theorem 1.1 of Huang-Wang-Wang <span><span>[20]</span></span> to any short time. As another byproduct, we extend the spectral inequality of Lebeau-Moyano <span><span>[27]</span></span> for real-analytic potentials to bounded continuous potentials in the one-dimensional case.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110695"},"PeriodicalIF":1.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312347","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 the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi equations","authors":"Marco Cirant","doi":"10.1016/j.jfa.2024.110692","DOIUrl":"10.1016/j.jfa.2024.110692","url":null,"abstract":"<div><div>We show in this paper that maximal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic <em>γ</em>-growth in the gradient holds in the full range <span><math><mi>q</mi><mo>&gt;</mo><mo>(</mo><mi>N</mi><mo>+</mo><mn>2</mn><mo>)</mo><mfrac><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>γ</mi></mrow></mfrac></math></span>. Our approach is based on new <span><math><mfrac><mrow><mi>γ</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span>-Hölder estimates, which are consequence of the decay at small scales of suitable nonlinear space and time Hölder quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110692"},"PeriodicalIF":1.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S002212362400380X/pdfft?md5=9f67759f78f3d63a96e6edeef4bf4034&pid=1-s2.0-S002212362400380X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312346","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":"Approximation of SBV functions with possibly infinite jump set","authors":"Sergio Conti ,&nbsp;Matteo Focardi ,&nbsp;Flaviana Iurlano","doi":"10.1016/j.jfa.2024.110686","DOIUrl":"10.1016/j.jfa.2024.110686","url":null,"abstract":"<div><div>We prove an approximation result for functions <span><math><mi>u</mi><mo>∈</mo><mi>S</mi><mi>B</mi><mi>V</mi><mo>(</mo><mi>Ω</mi><mo>;</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></math></span> such that ∇<em>u</em> is <em>p</em>-integrable, <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo>&lt;</mo><mo>∞</mo></math></span>, and <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mo>|</mo><mo>[</mo><mi>u</mi><mo>]</mo><mo>|</mo><mo>)</mo></math></span> is integrable over the jump set (whose <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> measure is possibly infinite), for some continuous, nondecreasing, subadditive function <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, with <span><math><msubsup><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>. The approximating functions <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> are piecewise affine with piecewise affine jump set; the convergence is that of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> for <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> and the convergence in energy for <span><math><mo>|</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup></math></span> and <span><math><mi>g</mi><mo>(</mo><mo>[</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>]</mo><mo>,</mo><msub><mrow><mi>ν</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo>)</mo></math></span> for suitable functions <em>g</em>. In particular, <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> converges to <em>u BV</em>-strictly, area-strictly, and strongly in <em>BV</em> after composition with a bilipschitz map. If in addition <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>)</mo><mo>&lt;</mo><mo>∞</mo></math></span>, we also have convergence of <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo>)</mo></math></span> to <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110686"},"PeriodicalIF":1.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326701","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}