{"title":"Periodic space-time homogenisation of the ϕ24 equation","authors":"Martin Hairer , Harprit Singh","doi":"10.1016/j.jfa.2024.110762","DOIUrl":"10.1016/j.jfa.2024.110762","url":null,"abstract":"<div><div>We consider the homogenisation problem for the <span><math><msubsup><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>4</mn></mrow></msubsup></math></span> equation on the torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, namely the behaviour as <span><math><mi>ε</mi><mo>→</mo><mn>0</mn></math></span> of the solutions to the equation <em>suggestively</em> written as<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>−</mo><mi>∇</mi><mo>⋅</mo><mi>A</mi><mo>(</mo><mi>x</mi><mo>/</mo><mi>ε</mi><mo>,</mo><mi>t</mi><mo>/</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>=</mo><mo>−</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>ξ</mi></math></span></span></span> where <em>ξ</em> denotes space-time white noise and <span><math><mi>A</mi><mo>:</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mi>R</mi></math></span> is uniformly elliptic, periodic and Hölder continuous. When the noise is regularised at scale <span><math><mi>δ</mi><mo>≪</mo><mn>1</mn></math></span> we show that any joint limit <span><math><mi>ε</mi><mo>,</mo><mi>δ</mi><mo>→</mo><mn>0</mn></math></span> recovers the classical dynamical <span><math><msubsup><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>4</mn></mrow></msubsup></math></span> model. In certain regimes or if the regularisation is chosen in a specific way adapted to the problem, we show that the counterterms can be chosen as explicit local functions of <em>A</em>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110762"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128549","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 simpliciality of function spaces not containing constants","authors":"Ondřej F.K. Kalenda, Jiří Spurný","doi":"10.1016/j.jfa.2024.110756","DOIUrl":"10.1016/j.jfa.2024.110756","url":null,"abstract":"<div><div>We investigate simpliciality of function spaces without constants. We prove, in particular, that several properties characterizing simpliciality in the classical case differ in this new setting. We also show that it may happen that a given point is not represented by any measure pseudosupported by the Choquet boundary, illustrating so limits of possible generalizations of the representation theorem. Moreover, we address the abstract Dirichlet problem in the new setting and establish some common points and nontrivial differences with the classical case.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110756"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743624","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 Leray transform: Distinguished measures, symmetries and polygamma inequalities","authors":"Luke D. Edholm , Yonatan Shelah","doi":"10.1016/j.jfa.2024.110746","DOIUrl":"10.1016/j.jfa.2024.110746","url":null,"abstract":"<div><div>New symmetries, norm computations and spectral information are obtained for the Leray transform on a class of unbounded hypersurfaces in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Emphasis is placed on certain distinguished measures, with results on operator norm monotonicity established by proving new polygamma inequalities. Classical techniques of Bernstein-Widder and Euler-Maclaurin play crucial roles in our analysis. Underpinning this work is a projective geometric theory of duality, which manifests here in the form of Hölder invariance.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110746"},"PeriodicalIF":1.7,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659809","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":"Power boundedness and related properties for weighted composition operators on S(Rd)","authors":"Vicente Asensio , Enrique Jordá , Thomas Kalmes","doi":"10.1016/j.jfa.2024.110745","DOIUrl":"10.1016/j.jfa.2024.110745","url":null,"abstract":"<div><div>We characterize those pairs <span><math><mo>(</mo><mi>ψ</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span> of smooth mappings <span><math><mi>ψ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><mi>C</mi><mo>,</mo><mi>φ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for which the corresponding weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>f</mi><mo>=</mo><mi>ψ</mi><mo>⋅</mo><mo>(</mo><mi>f</mi><mo>∘</mo><mi>φ</mi><mo>)</mo></math></span> acts continuously on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in terms of <span><math><mi>ψ</mi><mo>,</mo><mi>φ</mi></math></span>. Among other things, as an application of our results we show that for a univariate polynomial <em>φ</em> with <span><math><mtext>deg</mtext><mo>(</mo><mi>φ</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span>, power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for every <span><math><mi>ψ</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> only depends on <em>φ</em> and that in this case power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> is equivalent to <span><math><msub><mrow><mo>(</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> converging to 0 in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></math></span> as well as to the uniform mean ergodicity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span>. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for which neither the multiplication operator <span><math><mi>f</mi><mo>↦</mo><mi>ψ</mi><mi>f</mi></math></span> nor the composition","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110745"},"PeriodicalIF":1.7,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659891","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":"Optimal bounds for the Dunkl kernel in the dihedral case","authors":"Jean-Philippe Anker , Bartosz Trojan","doi":"10.1016/j.jfa.2024.110743","DOIUrl":"10.1016/j.jfa.2024.110743","url":null,"abstract":"<div><div>We establish sharp upper and lower estimates of the Dunkl kernel in the case of dihedral groups.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110743"},"PeriodicalIF":1.7,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659808","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":"Scalar curvature rigidity and the higher mapping degree","authors":"Thomas Tony","doi":"10.1016/j.jfa.2024.110744","DOIUrl":"10.1016/j.jfa.2024.110744","url":null,"abstract":"<div><div>A closed connected oriented Riemannian manifold <em>N</em> with non-vanishing Euler characteristic, non-negative curvature operator and <span><math><mn>0</mn><mo><</mo><mn>2</mn><msub><mrow><mi>Ric</mi></mrow><mrow><mi>N</mi></mrow></msub><mo><</mo><msub><mrow><mi>scal</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> is area-rigid in the sense that any area non-increasing spin map <span><math><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></math></span> with non-vanishing <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-degree and <span><math><msub><mrow><mi>scal</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>≥</mo><msub><mrow><mi>scal</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>∘</mo><mi>f</mi></math></span> is a Riemannian submersion with <span><math><msub><mrow><mi>scal</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>=</mo><msub><mrow><mi>scal</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>∘</mo><mi>f</mi></math></span>. This is due to Goette and Semmelmann and generalizes a result by Llarull. In this article, we show area-rigidity for not necessarily orientable manifolds with respect to a larger class of maps <span><math><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></math></span> by replacing the topological condition on the <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-degree by a less restrictive condition involving the so-called higher mapping degree. This includes fiber bundles over even dimensional spheres with enlargeable fibers, e.g. <span><math><msub><mrow><mi>pr</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>:</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>→</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>. We develop a technique to extract from a non-vanishing higher index a geometrically useful family of almost <figure><img></figure>-harmonic sections. This also leads to a new proof of the fact that any closed connected spin manifold with non-negative scalar curvature and non-trivial Rosenberg index is Ricci flat.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110744"},"PeriodicalIF":1.7,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659890","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":"C⁎-algebras associated to directed graphs of groups, and models of Kirchberg algebras","authors":"Victor Wu","doi":"10.1016/j.jfa.2024.110740","DOIUrl":"10.1016/j.jfa.2024.110740","url":null,"abstract":"<div><div>We introduce <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras associated to directed graphs of groups. In particular, we associate a combinatorial <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra to each row-finite directed graph of groups with no sources, and show that this <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110740"},"PeriodicalIF":1.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659888","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":"Pure ⁎-homomorphisms","authors":"Joan Bosa , Eduard Vilalta","doi":"10.1016/j.jfa.2024.110739","DOIUrl":"10.1016/j.jfa.2024.110739","url":null,"abstract":"<div><div>We introduce and study the notion of pureness for *-homomorphisms and, more generally, for cpc order-zero maps. After providing various important examples of pureness, we show our main result: Any composition of two pure maps factors through a pure object up to Cuntz equivalence. This is used to obtain several factorization results at the level of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110739"},"PeriodicalIF":1.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659934","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":"Multi-window STFT phase retrieval: Lattice uniqueness","authors":"Philipp Grohs , Lukas Liehr , Martin Rathmair","doi":"10.1016/j.jfa.2024.110733","DOIUrl":"10.1016/j.jfa.2024.110733","url":null,"abstract":"<div><div>Short-time Fourier transform (STFT) phase retrieval refers to the reconstruction of a function <em>f</em> from its spectrogram, i.e., the magnitudes of its short-time Fourier transform <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi></math></span> with window function <em>g</em>. While it is known that for appropriate windows, any function <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> can be reconstructed from the full spectrogram <span><math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo></math></span>, in practical scenarios, the reconstruction must be achieved from discrete samples, typically taken on a lattice. It turns out that the sampled problem becomes much more subtle: recent results have demonstrated that uniqueness via lattice-sampling is unachievable, irrespective of the choice of the window function or the lattice density. In the present paper, we initiate the study of multi-window STFT phase retrieval as a way to effectively bypass the discretization barriers encountered in the single-window case. By establishing a link between multi-window Gabor systems, sampling in Fock space, and phase retrieval for finite frames, we derive conditions under which square-integrable functions can be uniquely recovered from spectrogram samples on a lattice. Specifically, we provide conditions on window functions <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, such that every <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is determined up to a global phase from<span><span><span><math><mrow><mo>(</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mi>f</mi><mo>(</mo><mi>A</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo><mo>,</mo><mspace></mspace><mo>…</mo><mo>,</mo><mspace></mspace><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msub><mi>f</mi><mo>(</mo><mi>A</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo><mo>)</mo></mrow></math></span></span></span> whenever <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> satisfies the density condition <span><math><mo>|</mo><mi>det</mi><mo></mo><mi>A</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≥</mo><mn>4</mn></math></span>. For real","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110733"},"PeriodicalIF":1.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659892","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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