Journal of Functional Analysis最新文献

{"title":"Continuous dependence on initial data for the solutions of 3-D anisotropic Navier-Stokes equations","authors":"Ping Zhang ,&nbsp;Weipeng Zhu","doi":"10.1016/j.jfa.2024.110689","DOIUrl":"10.1016/j.jfa.2024.110689","url":null,"abstract":"<div><div>In this paper, we prove the continuous dependence on the initial data for the solutions of 3-D incompressible anisotropic Navier-Stokes equations in the functional space <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo><mover><mo>=</mo><mrow><mi>def</mi></mrow></mover><mo>{</mo><mi>u</mi><mo>:</mo><mi>u</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>;</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>s</mi></mrow></msup><mo>)</mo><mspace></mspace><mtext>with</mtext><mspace></mspace><msub><mrow><mi>∇</mi></mrow><mrow><mi>h</mi></mrow></msub><mi>u</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mo>]</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>[</mo><mo>;</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>s</mi></mrow></msup><mo>)</mo><mspace></mspace><mo>}</mo></math></span> for <span><math><mi>s</mi><mo>&gt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. We also show the non-uniform continuity of the data-to-solution map in <span><math><mi>C</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>;</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>s</mi></mrow></msup><mo>)</mo></math></span> for <span><math><mi>s</mi><mo>&gt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, which makes sharp contrast with the corresponding result for the classical 3-D Navier-Stokes equations.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110689"},"PeriodicalIF":1.7,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312832","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":"Douglas-Rudin approximation theorem for operator-valued functions on the unit ball of Cd","authors":"Poornendu Kumar ,&nbsp;Shubham Rastogi ,&nbsp;Raghavendra Tripathi","doi":"10.1016/j.jfa.2024.110685","DOIUrl":"10.1016/j.jfa.2024.110685","url":null,"abstract":"<div><div>Douglas and Rudin proved that any unimodular function on the unit circle <span><math><mi>T</mi></math></span> can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result (2009) <span><span>[4]</span></span> on the approximation of matrix-valued unimodular functions on <span><math><mi>T</mi></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110685"},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312834","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":"Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data","authors":"Andrew J. Morris,&nbsp;Andrew J. Turner","doi":"10.1016/j.jfa.2024.110680","DOIUrl":"10.1016/j.jfa.2024.110680","url":null,"abstract":"<div><div>We develop a holomorphic functional calculus for first-order operators <em>DB</em> to solve boundary value problems for Schrödinger equations <span><math><mo>−</mo><mi>div</mi><mspace></mspace><mi>A</mi><mi>∇</mi><mi>u</mi><mo>+</mo><mi>a</mi><mi>V</mi><mi>u</mi><mo>=</mo><mn>0</mn></math></span> in the upper half-space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span> with <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. This relies on quadratic estimates for <em>DB</em>, which are proved for coefficients <span><math><mi>A</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>V</mi></math></span> that are independent of the transversal direction to the boundary, and comprised of a complex-elliptic pair <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>a</mi><mo>)</mo></math></span> that are bounded and measurable, and a singular potential <em>V</em> in either <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> or the reverse Hölder class <span><math><msup><mrow><mi>B</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> with <span><math><mi>q</mi><mo>≥</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn><mo>}</mo></math></span>. In the latter case, square function bounds are also shown to be equivalent to non-tangential maximal function bounds. This allows us to prove that the (Dirichlet) Regularity and Neumann boundary value problems with <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>-data are well-posed if and only if certain boundary trace operators defined by the functional calculus are isomorphisms. We prove this property when the principal coefficient matrix <em>A</em> has either a Hermitian or block structure. More generally, the set of all complex coefficients for which the boundary value problems are well-posed is shown to be open.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110680"},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003689/pdfft?md5=6d3edebd34056c1c291ac6d370cafe17&pid=1-s2.0-S0022123624003689-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312836","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":"Classification of equivariantly O2-stable amenable actions on nuclear C⁎-algebras","authors":"Matteo Pagliero,&nbsp;Gábor Szabó","doi":"10.1016/j.jfa.2024.110683","DOIUrl":"10.1016/j.jfa.2024.110683","url":null,"abstract":"<div><div>Given a second-countable, locally compact group <em>G</em>, we consider amenable <em>G</em>-actions on separable, stable, nuclear <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras that are isometrically shift-absorbing and tensorially absorb the trivial action on the Cuntz algebra <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. We show that such actions are classified up to cocycle conjugacy by the induced <em>G</em>-action on the primitive ideal space. In the special case when <em>G</em> is exact, we prove a unital version of our classification theorem. For compact groups, we obtain a classification up to conjugacy.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 2","pages":"Article 110683"},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319597","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":"High moments of the SHE in the clustering regimes","authors":"Li-Cheng Tsai","doi":"10.1016/j.jfa.2024.110675","DOIUrl":"10.1016/j.jfa.2024.110675","url":null,"abstract":"<div><div>We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under the delta(-like) initial condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar–Parisi–Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper <span><span>[75]</span></span> to prove an <em>n</em>-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110675"},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312833","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":"Singular extension of critical Sobolev mappings under an exponential weak-type estimate","authors":"Bohdan Bulanyi ,&nbsp;Jean Van Schaftingen","doi":"10.1016/j.jfa.2024.110681","DOIUrl":"10.1016/j.jfa.2024.110681","url":null,"abstract":"<div><div>Given <span><math><mi>m</mi><mo>∈</mo><mi>N</mi><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> and a compact Riemannian manifold <span><math><mi>N</mi></math></span>, we construct for every map <em>u</em> in the critical Sobolev space <span><math><msup><mrow><mi>W</mi></mrow><mrow><mi>m</mi><mo>/</mo><mo>(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>,</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mi>N</mi><mo>)</mo></math></span>, a map <span><math><mi>U</mi><mo>:</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>→</mo><mi>N</mi></math></span> whose trace is <em>u</em> and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110681"},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142320113","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":"Higher index theory for spaces with an FCE-by-FCE structure","authors":"Jintao Deng ,&nbsp;Liang Guo ,&nbsp;Qin Wang ,&nbsp;Guoliang Yu","doi":"10.1016/j.jfa.2024.110679","DOIUrl":"10.1016/j.jfa.2024.110679","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mo>(</mo><mn>1</mn><mo>→</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> be a sequence of extensions of finite groups. Assume that the coarse disjoint unions of <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> and <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> have bounded geometry. The sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> is said to have an <em>FCE-by-FCE structure</em>, if the sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> and the sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> admit <em>a fibred coarse embedding</em> into Hilbert space. In this paper, we prove the coarse Novikov conjecture holds for the sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> with an FCE-by-FCE structure.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110679"},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274603","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":"Density of compactly supported smooth functions CC∞(Rd) in Musielak-Orlicz-Sobolev spaces W1,Φ(Ω)","authors":"Anna Kamińska ,&nbsp;Mariusz Żyluk","doi":"10.1016/j.jfa.2024.110677","DOIUrl":"10.1016/j.jfa.2024.110677","url":null,"abstract":"<div><div>We investigate here the density of the set of the restrictions from <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> to <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> in the Musielak-Orlicz-Sobolev space <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>Φ</mi></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>. It is a continuation of article <span><span>[15]</span></span>, where we have studied density of <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in <span><math><msup><mrow><mi>W</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>Φ</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> for <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span>. The main theorem states that for an open subset <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with its boundary of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, and Musielak-Orlicz function Φ satisfying condition (A1) which is a sort of log-Hölder continuity and the growth condition <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, the set of restrictions of functions from <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> to Ω is dense in <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>Φ</mi></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>. We obtain a corresponding result in variable exponent Sobolev space <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> under the assumption that the exponent <span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is essentially bounded on Ω and <span><math><mi>Φ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msup></math></span>, <span><math><mi>t</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><mi>x</mi><mo>∈</mo><mi>Ω</mi></math></span>, satisfies the log-Hölder condition.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110677"},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312837","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":"An integrable bound for rough stochastic partial differential equations with applications to invariant manifolds and stability","authors":"M. Ghani Varzaneh,&nbsp;S. Riedel","doi":"10.1016/j.jfa.2024.110676","DOIUrl":"10.1016/j.jfa.2024.110676","url":null,"abstract":"<div><p>We study semilinear rough stochastic partial differential equations as introduced in Gerasimovičs and Hairer (2019) <span><span>[31]</span></span>. We provide <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>-integrable a priori bounds for the solution and its linearization in case the equation is driven by a suitable Gaussian process. Using the multiplicative ergodic theorem for Banach spaces, we can deduce the existence of a Lyapunov spectrum for the linearized equation around stationary points. The existence of local stable, unstable, and center manifolds around stationary points is provided. In the case where all Lyapunov exponents are negative, local exponential stability can be deduced. We illustrate our findings with several examples.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110676"},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003641/pdfft?md5=b80218becb1906e603d5ede602597273&pid=1-s2.0-S0022123624003641-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274600","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":"A probabilistic approach to Lorentz balls ℓq,1n","authors":"Zakhar Kabluchko ,&nbsp;Joscha Prochno ,&nbsp;Mathias Sonnleitner","doi":"10.1016/j.jfa.2024.110682","DOIUrl":"10.1016/j.jfa.2024.110682","url":null,"abstract":"<div><p>We develop a probabilistic approach to study the volumetric and geometric properties of unit balls <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>q</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> of finite-dimensional Lorentz sequence spaces <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mi>q</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>. More precisely, we show that the empirical distribution of a random vector <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> uniformly distributed on its volume normalized unit ball converges weakly to a compactly supported symmetric probability distribution with explicitly given density; as a consequence we obtain a weak Poincaré-Maxwell-Borel principle for any fixed number <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span> of coordinates of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. Moreover, we prove a central limit theorem for the largest coordinate of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span>, demonstrating a quite different behavior than in the case of the <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> balls, where a Gumbel distribution appears in the limit. Finally, we prove a Schechtman-Schmuckenschläger type result for the asymptotic volume of intersections of volume normalized <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mi>q</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> balls.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110682"},"PeriodicalIF":1.7,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003707/pdfft?md5=9e7a15addc6eca991b6cc2bfa89d8f84&pid=1-s2.0-S0022123624003707-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274601","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}