Pub Date : 2024-12-06DOI: 10.1016/j.jfa.2024.110790
Cédric Arhancet
We investigate a new class of unital quantum channels on , acting as radial multipliers when we identify the matrix algebra with a finite-dimensional fermion algebra. Our primary contribution lies in the precise computation of the (optimal) rate at which classical information can be transmitted through these channels from a sender to a receiver when they share an unlimited amount of entanglement. Our approach relies on new connections between fermion algebras with the n-dimensional discrete hypercube . Significantly, our calculations yield exact values applicable to the operators of the fermionic Ornstein-Uhlenbeck semigroup. This advancement not only provides deeper insights into the structure and behaviour of these channels but also enhances our understanding of Quantum Information Theory in a dimension-independent context.
{"title":"Entanglement-assisted classical capacities of some channels acting as radial multipliers on fermion algebras","authors":"Cédric Arhancet","doi":"10.1016/j.jfa.2024.110790","DOIUrl":"10.1016/j.jfa.2024.110790","url":null,"abstract":"<div><div>We investigate a new class of unital quantum channels on <span><math><msub><mrow><mi>M</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></msub></math></span>, acting as radial multipliers when we identify the matrix algebra <span><math><msub><mrow><mi>M</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></msub></math></span> with a finite-dimensional fermion algebra. Our primary contribution lies in the precise computation of the (optimal) rate at which classical information can be transmitted through these channels from a sender to a receiver when they share an unlimited amount of entanglement. Our approach relies on new connections between fermion algebras with the <em>n</em>-dimensional discrete hypercube <span><math><msup><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Significantly, our calculations yield exact values applicable to the operators of the fermionic Ornstein-Uhlenbeck semigroup. This advancement not only provides deeper insights into the structure and behaviour of these channels but also enhances our understanding of Quantum Information Theory in a dimension-independent context.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110790"},"PeriodicalIF":1.7,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092983","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}
Pub Date : 2024-12-06DOI: 10.1016/j.jfa.2024.110786
Siming He
We consider the three-dimensional parabolic-parabolic Patlak-Keller-Segel equations (PKS) subject to ambient flows. Without the ambient fluid flow, the equation is super-critical in three-dimension and has finite-time blow-up solutions with arbitrarily small -mass. In this study, we show that a family of time-dependent alternating shear flows, inspired by the clever ideas of Tarek Elgindi [39], can suppress the chemotactic blow-up in these systems.
{"title":"Time-dependent flows and their applications in parabolic-parabolic Patlak-Keller-Segel systems Part I: Alternating flows","authors":"Siming He","doi":"10.1016/j.jfa.2024.110786","DOIUrl":"10.1016/j.jfa.2024.110786","url":null,"abstract":"<div><div>We consider the three-dimensional parabolic-parabolic Patlak-Keller-Segel equations (PKS) subject to ambient flows. Without the ambient fluid flow, the equation is super-critical in three-dimension and has finite-time blow-up solutions with arbitrarily small <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-mass. In this study, we show that a family of time-dependent alternating shear flows, inspired by the clever ideas of Tarek Elgindi <span><span>[39]</span></span>, can suppress the chemotactic blow-up in these systems.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110786"},"PeriodicalIF":1.7,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128207","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}
Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110761
Pierre Gilles Lemarié-Rieusset
We construct non-trivial steady solutions in for the 2D Navier–Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to introduce a notion of special (non square integrable) solutions.
{"title":"Highly singular (frequentially sparse) steady solutions for the 2D Navier–Stokes equations on the torus","authors":"Pierre Gilles Lemarié-Rieusset","doi":"10.1016/j.jfa.2024.110761","DOIUrl":"10.1016/j.jfa.2024.110761","url":null,"abstract":"<div><div>We construct non-trivial steady solutions in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> for the 2D Navier–Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to introduce a notion of special (non square integrable) solutions.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110761"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743690","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}
Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110760
Filippo Boni , Simone Dovetta , Enrico Serra
We investigate the existence of normalized ground states for Schrödinger equations on noncompact metric graphs in presence of nonlinear point defects, described by nonlinear δ-interactions at some of the vertices of the graph. For graphs with finitely many vertices, we show that ground states exist for every mass and every -subcritical power. For graphs with infinitely many vertices, we focus on periodic graphs and, in particular, on -periodic graphs and on a prototypical -periodic graph, the two–dimensional square grid. We provide a set of results unravelling nontrivial threshold phenomena both on the mass and on the nonlinearity power, showing the strong dependence of the ground state problem on the interplay between the degree of periodicity of the graph, the total number of point defects and their dislocation in the graph.
{"title":"Normalized ground states for Schrödinger equations on metric graphs with nonlinear point defects","authors":"Filippo Boni , Simone Dovetta , Enrico Serra","doi":"10.1016/j.jfa.2024.110760","DOIUrl":"10.1016/j.jfa.2024.110760","url":null,"abstract":"<div><div>We investigate the existence of normalized ground states for Schrödinger equations on noncompact metric graphs in presence of nonlinear point defects, described by nonlinear <em>δ</em>-interactions at some of the vertices of the graph. For graphs with finitely many vertices, we show that ground states exist for every mass and every <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-subcritical power. For graphs with infinitely many vertices, we focus on periodic graphs and, in particular, on <span><math><mi>Z</mi></math></span>-periodic graphs and on a prototypical <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-periodic graph, the two–dimensional square grid. We provide a set of results unravelling nontrivial threshold phenomena both on the mass and on the nonlinearity power, showing the strong dependence of the ground state problem on the interplay between the degree of periodicity of the graph, the total number of point defects and their dislocation in the graph.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110760"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743622","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}
Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110755
Hendrik De Bie , Pan Lian , Frederick Maes
In this paper, we study the pointwise bounds for the kernel of the -generalized Fourier transform with , introduced by Ben Saïd, Kobayashi and Ørsted. We present explicit formulas for the case , which show that the kernels can exhibit polynomial growth. Subsequently, we provide a polynomial bound for the even dimensional kernel for this transform, focusing on the cases with finite order. Furthermore, by utilizing an estimation for the Prabhakar function, it is found that the -generalized Fourier kernel is bounded by a constant when and , except within an angular domain that diminishes as . As a byproduct, we prove that the -generalized Fourier kernel is uniformly bounded, when and .
在本文中，我们研究了Ben Saïd， Kobayashi和Ørsted引入的(κ,a)-广义傅里叶变换（κ≡0）核的点向界。我们给出了a=4情况下的显式公式，表明核可以呈现多项式增长。随后，我们给出了该变换的偶维核的多项式界，重点讨论了有限阶的情况。进一步，通过对Prabhakar函数的估计，我们发现(0,a)-广义傅里叶核在a>；1和m≥2时被一个常数限定，除了在角域内随着a→∞而减小。作为副产物，我们证明了(0,2 r /n)-广义傅里叶核是一致有界的，当m=2且r，n∈n。
{"title":"Bounds for the kernel of the (κ,a)-generalized Fourier transform","authors":"Hendrik De Bie , Pan Lian , Frederick Maes","doi":"10.1016/j.jfa.2024.110755","DOIUrl":"10.1016/j.jfa.2024.110755","url":null,"abstract":"<div><div>In this paper, we study the pointwise bounds for the kernel of the <span><math><mo>(</mo><mi>κ</mi><mo>,</mo><mi>a</mi><mo>)</mo></math></span>-generalized Fourier transform with <span><math><mi>κ</mi><mo>≡</mo><mn>0</mn></math></span>, introduced by Ben Saïd, Kobayashi and Ørsted. We present explicit formulas for the case <span><math><mi>a</mi><mo>=</mo><mn>4</mn></math></span>, which show that the kernels can exhibit polynomial growth. Subsequently, we provide a polynomial bound for the even dimensional kernel for this transform, focusing on the cases with finite order. Furthermore, by utilizing an estimation for the Prabhakar function, it is found that the <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mi>a</mi><mo>)</mo></math></span>-generalized Fourier kernel is bounded by a constant when <span><math><mi>a</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>, except within an angular domain that diminishes as <span><math><mi>a</mi><mo>→</mo><mo>∞</mo></math></span>. As a byproduct, we prove that the <span><math><mo>(</mo><mn>0</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>ℓ</mi></mrow></msup><mo>/</mo><mi>n</mi><mo>)</mo></math></span>-generalized Fourier kernel is uniformly bounded, when <span><math><mi>m</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>ℓ</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110755"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743689","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}
Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110754
Teun van Nuland , Fedor Sukochev , Dmitriy Zanin
We explicitly compute the local invariants (heat kernel coefficients) of a conformally deformed non-commutative d-torus using multiple operator integrals. We derive a recursive formula that easily produces an explicit expression for the local invariants of any order k and in any dimension d. Our recursive formula can conveniently produce all formulas related to the modular operator, which before were obtained in incremental steps for and . We exemplify this by writing down some known (, ) and some novel (, ) formulas in the modular operator.
{"title":"Local invariants of conformally deformed non-commutative tori II: Multiple operator integrals","authors":"Teun van Nuland , Fedor Sukochev , Dmitriy Zanin","doi":"10.1016/j.jfa.2024.110754","DOIUrl":"10.1016/j.jfa.2024.110754","url":null,"abstract":"<div><div>We explicitly compute the local invariants (heat kernel coefficients) of a conformally deformed non-commutative <em>d</em>-torus using multiple operator integrals. We derive a recursive formula that easily produces an explicit expression for the local invariants of any order <em>k</em> and in any dimension <em>d</em>. Our recursive formula can conveniently produce all formulas related to the modular operator, which before were obtained in incremental steps for <span><math><mi>d</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></math></span> and <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>}</mo></math></span>. We exemplify this by writing down some known (<span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>) and some novel (<span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>) formulas in the modular operator.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110754"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178223","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}
Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110758
Panu Lahti
We investigate a version of Alberti's rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual steps, but the most crucial step consists in showing for that at -a.e. , the mapping f “behaves non-quasiconformally”.
{"title":"Alberti's rank one theorem and quasiconformal mappings in metric measure spaces","authors":"Panu Lahti","doi":"10.1016/j.jfa.2024.110758","DOIUrl":"10.1016/j.jfa.2024.110758","url":null,"abstract":"<div><div>We investigate a version of Alberti's rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual steps, but the most crucial step consists in showing for <span><math><mi>f</mi><mo>∈</mo><mrow><mi>BV</mi></mrow><mo>(</mo><mi>X</mi><mo>;</mo><mi>Y</mi><mo>)</mo></math></span> that at <span><math><msup><mrow><mo>‖</mo><mi>D</mi><mi>f</mi><mo>‖</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span>-a.e. <span><math><mi>x</mi><mo>∈</mo><mi>X</mi></math></span>, the mapping <em>f</em> “behaves non-quasiconformally”.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110758"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743623","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}
Pub Date : 2024-11-20DOI: 10.1016/j.jfa.2024.110753
Kevin O'Neill
The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set , how can we tell if there exists a d-dimensional, -smooth manifold ? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like that used in the solution of the classical Whitney extension problem and a topological condition. This condition is the existence of a continuous selection for Grassmannian-valued functions, meant to reflect the collection of possible tangent spaces. We demonstrate the necessity of this condition in general and its non-redundancy in an example, while also showing it need not be checked when .
{"title":"A Whitney extension problem for manifolds","authors":"Kevin O'Neill","doi":"10.1016/j.jfa.2024.110753","DOIUrl":"10.1016/j.jfa.2024.110753","url":null,"abstract":"<div><div>The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set <span><math><mi>E</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, how can we tell if there exists a <em>d</em>-dimensional, <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>-smooth manifold <span><math><mi>M</mi><mo>⊃</mo><mi>E</mi></math></span>? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like that used in the solution of the classical Whitney extension problem and a topological condition. This condition is the existence of a continuous selection for Grassmannian-valued functions, meant to reflect the collection of possible tangent spaces. We demonstrate the necessity of this condition in general and its non-redundancy in an example, while also showing it need not be checked when <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110753"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143093263","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}
Pub Date : 2024-11-20DOI: 10.1016/j.jfa.2024.110763
A. Fovelle
We prove an optimal result of stability under -sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property.
{"title":"Asymptotic smoothness, concentration properties in Banach spaces and applications","authors":"A. Fovelle","doi":"10.1016/j.jfa.2024.110763","DOIUrl":"10.1016/j.jfa.2024.110763","url":null,"abstract":"<div><div>We prove an optimal result of stability under <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110763"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743621","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}
Pub Date : 2024-11-20DOI: 10.1016/j.jfa.2024.110757
Damian Głodkowski , Agnieszka Widz
We define a σ-centered notion of forcing that forces the existence of a Boolean algebra with the Grothendieck property and without the Nikodym property. In particular, the existence of such an algebra is consistent with the negation of the continuum hypothesis. The algebra we construct consists of Borel subsets of the Cantor set and has cardinality . We also show how to apply our method to streamline Talagrand's construction of such an algebra under the continuum hypothesis.
{"title":"Epic math battle of history: Grothendieck vs Nikodym","authors":"Damian Głodkowski , Agnieszka Widz","doi":"10.1016/j.jfa.2024.110757","DOIUrl":"10.1016/j.jfa.2024.110757","url":null,"abstract":"<div><div>We define a <em>σ</em>-centered notion of forcing that forces the existence of a Boolean algebra with the Grothendieck property and without the Nikodym property. In particular, the existence of such an algebra is consistent with the negation of the continuum hypothesis. The algebra we construct consists of Borel subsets of the Cantor set and has cardinality <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. We also show how to apply our method to streamline Talagrand's construction of such an algebra under the continuum hypothesis.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110757"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743589","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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