Journal of Functional Analysis最新文献

{"title":"Quantitative observability for one-dimensional Schrödinger equations with potentials","authors":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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":"Sobolev smoothing estimates for bilinear maximal operators with fractal dilation sets","authors":"","doi":"10.1016/j.jfa.2024.110694","DOIUrl":"10.1016/j.jfa.2024.110694","url":null,"abstract":"<div><div>Given a hypersurface <span><math><mi>S</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>d</mi></mrow></msup></math></span>, we study the bilinear averaging operator that averages a pair of functions over <em>S</em>, as well as more general bilinear multipliers of limited decay and various maximal analogs. Of particular interest are bilinear maximal operators associated to a fractal dilation set <span><math><mi>E</mi><mo>⊂</mo><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span>; in this case, the boundedness region of the maximal operator is associated to the geometry of the hypersurface and various notions of the dimension of the dilation set. In particular, we determine Sobolev smoothing estimates at the exponent <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> using Fourier-analytic methods, which allow us to deduce additional <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> improving bounds for the operators and sparse bounds and their weighted corollaries for the associated multi-scale maximal functions. We also extend the method to study analogues of these questions for the triangle averaging operator and biparameter averaging operators. In addition, some necessary conditions for boundedness of these operators are obtained.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326702","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":"Approximation of SBV functions with possibly infinite jump set","authors":"","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":null,"pages":null},"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}

{"title":"Generalized hyperbolicity, stability and expansivity for operators on locally convex spaces","authors":"","doi":"10.1016/j.jfa.2024.110696","DOIUrl":"10.1016/j.jfa.2024.110696","url":null,"abstract":"<div><div>We introduce and study the notions of (generalized) hyperbolicity, topological stability and (uniform) topological expansivity for operators on locally convex spaces. We prove that every generalized hyperbolic operator on a locally convex space has the finite shadowing property. Contrary to what happens in the Banach space setting, hyperbolic operators on Fréchet spaces may fail to have the shadowing property, but we find additional conditions that ensure the validity of the shadowing property. Assuming that the space is sequentially complete, we prove that generalized hyperbolicity implies the strict periodic shadowing property, but we also show that the hypothesis of sequential completeness is essential. We show that operators with the periodic shadowing property on topological vector spaces have other interesting dynamical behaviors, including the fact that the restriction of such an operator to its chain recurrent set is topologically mixing and Devaney chaotic. We prove that topologically stable operators on locally convex spaces have the finite shadowing property and the strict periodic shadowing property. As a consequence, topologically stable operators on Banach spaces have the shadowing property. Moreover, we prove that generalized hyperbolicity implies topological stability for operators on Banach spaces. We prove that uniformly topologically expansive operators on locally convex spaces are neither Li-Yorke chaotic nor topologically transitive. Finally, we characterize the notion of topological expansivity for weighted shifts on Fréchet sequence spaces. Several examples are provided.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003847/pdfft?md5=e7519f57e3ee95a9d05222beccd3f5c5&pid=1-s2.0-S0022123624003847-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314783","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":"Statistical mechanics of the wave maps equation in dimension 1 + 1","authors":"","doi":"10.1016/j.jfa.2024.110688","DOIUrl":"10.1016/j.jfa.2024.110688","url":null,"abstract":"<div><div>We study wave maps with values in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, defined on the future light cone <span><math><mo>{</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>≤</mo><mi>t</mi><mo>}</mo><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></msup></math></span>, with prescribed data at the boundary <span><math><mo>{</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mi>t</mi><mo>}</mo></math></span>. Based on the work of Keel and Tao, we prove that the problem is well-posed for locally absolutely continuous boundary data. We design a discrete version of the problem and prove that for every absolutely continuous boundary data, the sequence of solutions of the discretised problem converges to the corresponding continuous wave map as the mesh size tends to 0.</div><div>Next, we consider the boundary data given by the <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-valued Brownian motion. We prove that the sequence of solutions of the discretised problems has a subsequence that converges in law in the topology of locally uniform convergence. We argue that the resulting random field can be interpreted as the wave-map evolution corresponding to the initial data given by the Gibbs distribution.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003768/pdfft?md5=d9593932c2fb61743a517d3804ef582f&pid=1-s2.0-S0022123624003768-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142316223","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":"Hypocoercivity in Hilbert spaces","authors":"","doi":"10.1016/j.jfa.2024.110691","DOIUrl":"10.1016/j.jfa.2024.110691","url":null,"abstract":"<div><div>The concept of hypocoercivity for linear evolution equations with dissipation is discussed and equivalent characterizations that were developed for the finite-dimensional case are extended to separable Hilbert spaces. Using the concept of a hypocoercivity index, quantitative estimates on the short-time and long-time decay behavior of a hypocoercive system are derived. As a useful tool for analyzing the structural properties, an infinite-dimensional staircase form is also derived and connections to linear systems and control theory are presented. Several examples illustrate the new concepts and the results are applied to the Lorentz kinetic equation.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003793/pdfft?md5=e2463e2bf7ceee0363d431fdcb34ac65&pid=1-s2.0-S0022123624003793-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314214","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":"Wolff potential estimates and Wiener criterion for nonlocal equations with Orlicz growth","authors":"","doi":"10.1016/j.jfa.2024.110690","DOIUrl":"10.1016/j.jfa.2024.110690","url":null,"abstract":"<div><div>We prove the Wolff potential estimates for nonlocal equations with Orlicz growth. As an application, we obtain the Wiener criterion in this framework, which provides a necessary and sufficient condition for boundary points to be regular. Our approach relies on the fine analysis of superharmonic functions in view of nonlocal nonlinear potential theory.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315675","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 elliptic maximal function","authors":"","doi":"10.1016/j.jfa.2024.110693","DOIUrl":"10.1016/j.jfa.2024.110693","url":null,"abstract":"<div><div>We study the elliptic maximal functions defined by averages over ellipses and rotated ellipses which are multi-parametric variants of the circular maximal function. We prove that those maximal functions are bounded on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> for some <span><math><mi>p</mi><mo>≠</mo><mo>∞</mo></math></span>. For this purpose, we obtain some sharp multi-parameter local smoothing estimates.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322339","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":"Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I","authors":"","doi":"10.1016/j.jfa.2024.110687","DOIUrl":"10.1016/j.jfa.2024.110687","url":null,"abstract":"<div><div>Let <em>G</em> be a (non compact) connected, simply connected, locally compact, second countable Lie group, either abelian or unimodular of type I, and let <em>ρ</em> be an irreducible unitary representation of <em>G</em>. Then, we define the analytic torsion of <em>G</em> localised at the representation <em>ρ</em>. The idea of considering localised invariants is due to Brodzki, Niblo, Plymen and Wright <span><span>[5]</span></span>, and was exploited in <span><span>[31]</span></span> to define a localised eta function. Next, let Γ be a discrete co compact subgroup of <em>G</em>. We use the localised analytic torsion to define the relative analytic torsion of the pair <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>Γ</mi><mo>)</mo></math></span>, and we prove that the last coincides with the Lott <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> analytic torsion of a covering space. We illustrate these constructions analysing in some details two examples: the abelian case, and the case <span><math><mi>G</mi><mo>=</mo><mi>H</mi></math></span>, the Heisenberg group.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003756/pdfft?md5=c742e607db225a998b538621bbeaade9&pid=1-s2.0-S0022123624003756-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314215","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}