Pub Date : 2024-08-31DOI: 10.1016/j.jfa.2024.110653
We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics equations in the half-plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform estimates of the error functions comes from the unboundedness of vorticity of strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential component of magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl boundary layer ansatz in sense in Sobolev framework. Compared with the homogeneous incompressible case considered in [33], there exists a strong boundary layer of density. Consequently, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.
{"title":"Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework","authors":"","doi":"10.1016/j.jfa.2024.110653","DOIUrl":"10.1016/j.jfa.2024.110653","url":null,"abstract":"<div><p>We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics equations in the half-plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> estimates of the error functions comes from the unboundedness of vorticity of strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential component of magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl boundary layer ansatz in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> sense in Sobolev framework. Compared with the homogeneous incompressible case considered in <span><span>[33]</span></span>, there exists a strong boundary layer of density. Consequently, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142157646","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 : 2024-08-31DOI: 10.1016/j.jfa.2024.110651
We prove the Widom–Sobolev formula for the asymptotic behaviour of truncated Wiener–Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum except when closing the asymptotics for twice differentiable test functions with Hölder singularities. The cut-off domains are allowed to have piecewise differentiable boundaries. In contrast to the case where the symbol is smooth in one variable, the resulting coefficient in the enhanced area law we obtain here remains as explicit for matrix-valued symbols as it is for scalar-valued symbols.
{"title":"The Widom–Sobolev formula for discontinuous matrix-valued symbols","authors":"","doi":"10.1016/j.jfa.2024.110651","DOIUrl":"10.1016/j.jfa.2024.110651","url":null,"abstract":"<div><p>We prove the Widom–Sobolev formula for the asymptotic behaviour of truncated Wiener–Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum except when closing the asymptotics for twice differentiable test functions with Hölder singularities. The cut-off domains are allowed to have piecewise differentiable boundaries. In contrast to the case where the symbol is smooth in one variable, the resulting coefficient in the enhanced area law we obtain here remains as explicit for matrix-valued symbols as it is for scalar-valued symbols.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003392/pdfft?md5=8010699661a21471a1a6bd426ef43d6a&pid=1-s2.0-S0022123624003392-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142161640","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 : 2024-08-31DOI: 10.1016/j.jfa.2024.110660
The goal of this work is to obtain (nearly) optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the limiting problem might not be differentiable on the entire space. Our main result is then to derive sharp rates of convergence in two distinct regimes. First, when the data is sufficiently regular, we obtain rates proportional to , with N being the number of particles, and we verify that is indeed optimal in this setting. Second, when the data is merely Lipschitz and semi-concave with respect to the first Wasserstein distance, we obtain rates proportional to . We do not expect this second estimate to be optimal, but it improves substantially on the existing literature. Moreover, we construct an example showing that the optimal rate is no faster than , and we conjecture that the optimal rate should indeed be exactly (at least when ). The key argument in our approach consists in mollifying the value function of the limiting problem in order to produce functions that are almost classical sub-solutions to the limiting Hamilton-Jacobi equation (which is a PDE set on the space of probability measures). These sub-solutions can be projected onto finite dimensional spaces and then compared with the value functions associated with the particle systems. In the end, this comparison is used to prove the most demanding bound in the estimates. The key challenge therein is thus to exhibit an appropriate form of mollification. We do so by employing sup-convolution within a convenient functional Hilbert space. To make the whole easier, we limit ourselves to the periodic setting. We also provide some examples to show that our results are sharp up to some extent.
{"title":"On the optimal rate for the convergence problem in mean field control","authors":"","doi":"10.1016/j.jfa.2024.110660","DOIUrl":"10.1016/j.jfa.2024.110660","url":null,"abstract":"<div><p>The goal of this work is to obtain (nearly) optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the limiting problem might not be differentiable on the entire space. Our main result is then to derive sharp rates of convergence in two distinct regimes. First, when the data is sufficiently regular, we obtain rates proportional to <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>, with <em>N</em> being the number of particles, and we verify that <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> is indeed optimal in this setting. Second, when the data is merely Lipschitz and semi-concave with respect to the first Wasserstein distance, we obtain rates proportional to <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>2</mn><mo>/</mo><mo>(</mo><mn>3</mn><mi>d</mi><mo>+</mo><mn>6</mn><mo>)</mo></mrow></msup></math></span>. We do not expect this second estimate to be optimal, but it improves substantially on the existing literature. Moreover, we construct an example showing that the optimal rate is no faster than <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup></math></span>, and we conjecture that the optimal rate should indeed be exactly <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup></math></span> (at least when <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>). The key argument in our approach consists in mollifying the value function of the limiting problem in order to produce functions that are almost classical sub-solutions to the limiting Hamilton-Jacobi equation (which is a PDE set on the space of probability measures). These sub-solutions can be projected onto finite dimensional spaces and then compared with the value functions associated with the particle systems. In the end, this comparison is used to prove the most demanding bound in the estimates. The key challenge therein is thus to exhibit an appropriate form of mollification. We do so by employing sup-convolution within a convenient functional Hilbert space. To make the whole easier, we limit ourselves to the periodic setting. We also provide some examples to show that our results are sharp up to some extent.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003483/pdfft?md5=5935f79205a900c37dc8ad1a51b88e6a&pid=1-s2.0-S0022123624003483-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151206","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 : 2024-08-31DOI: 10.1016/j.jfa.2024.110643
<div><p>We consider the Cauchy problem of the porous medium type reaction-diffusion equation<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>ρ</mi><mo>=</mo><mi>Δ</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mi>ρ</mi><mi>g</mi><mo>(</mo><mi>ρ</mi><mo>)</mo><mo>,</mo><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mspace></mspace><mi>m</mi><mo>></mo><mn>1</mn><mo>,</mo></math></span></span></span> where <em>g</em> is the given monotonic decreasing function with the density critical threshold <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> satisfying <span><math><mi>g</mi><mo>(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. We prove that the pressure <span><math><mi>P</mi><mo>:</mo><mo>=</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mfrac><msup><mrow><mi>ρ</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> in <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> tends to the pressure critical threshold <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mfrac><msup><mrow><mo>(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> at the time decay rate <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. If the initial density <span><math><mi>ρ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span> is compactly supported, we justify that the support <span><math><mo>{</mo><mi>x</mi><mo>:</mo><mi>ρ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>></mo><mn>0</mn><mo>}</mo></math></span> of the density <em>ρ</em> expands exponentially in time. Furthermore, we show that there exists a time <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></math></span> such that the pressure <em>P</em> is Lipschitz continuous for <span><math><mi>t</mi><mo>></mo><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, which is the optimal (sharp) regularity of the pressure, and the free surface <span><math><mo>∂</mo><mo>{</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>:</mo><mi>ρ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>></mo><mn>0</mn><mo>}<
{"title":"Porous medium type reaction-diffusion equation: Large time behaviors and regularity of free boundary","authors":"","doi":"10.1016/j.jfa.2024.110643","DOIUrl":"10.1016/j.jfa.2024.110643","url":null,"abstract":"<div><p>We consider the Cauchy problem of the porous medium type reaction-diffusion equation<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>ρ</mi><mo>=</mo><mi>Δ</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mi>ρ</mi><mi>g</mi><mo>(</mo><mi>ρ</mi><mo>)</mo><mo>,</mo><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mspace></mspace><mi>m</mi><mo>></mo><mn>1</mn><mo>,</mo></math></span></span></span> where <em>g</em> is the given monotonic decreasing function with the density critical threshold <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> satisfying <span><math><mi>g</mi><mo>(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. We prove that the pressure <span><math><mi>P</mi><mo>:</mo><mo>=</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mfrac><msup><mrow><mi>ρ</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> in <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> tends to the pressure critical threshold <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mfrac><msup><mrow><mo>(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> at the time decay rate <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. If the initial density <span><math><mi>ρ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span> is compactly supported, we justify that the support <span><math><mo>{</mo><mi>x</mi><mo>:</mo><mi>ρ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>></mo><mn>0</mn><mo>}</mo></math></span> of the density <em>ρ</em> expands exponentially in time. Furthermore, we show that there exists a time <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></math></span> such that the pressure <em>P</em> is Lipschitz continuous for <span><math><mi>t</mi><mo>></mo><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, which is the optimal (sharp) regularity of the pressure, and the free surface <span><math><mo>∂</mo><mo>{</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>:</mo><mi>ρ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>></mo><mn>0</mn><mo>}<","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151202","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 : 2024-08-31DOI: 10.1016/j.jfa.2024.110647
We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension N of the matrices grows to infinity, the operator norm of such polynomials q converges to a deterministic limit with a rate of convergence of . Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviors.
我们研究了多个独立维格纳矩阵的赫米提非交换二次多项式。我们证明,除了一些特定的可还原情况外,多项式的极限谱密度在其边缘总是有平方根增长,并证明了这些边缘周围的最优局部规律。结合这两个结果,我们确定,当矩阵的维数 N 增长到无穷大时,此类多项式 q 的算子规范会收敛到一个确定的极限,收敛速率为 N-2/3+o(1)。这里,收敛速率的指数是最优的。对于特定的可还原情况,我们还提供了所有可能的边缘行为分类。
{"title":"Norm convergence rate for multivariate quadratic polynomials of Wigner matrices","authors":"","doi":"10.1016/j.jfa.2024.110647","DOIUrl":"10.1016/j.jfa.2024.110647","url":null,"abstract":"<div><p>We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension <em>N</em> of the matrices grows to infinity, the operator norm of such polynomials <em>q</em> converges to a deterministic limit with a rate of convergence of <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>. Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviors.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003355/pdfft?md5=4e27857829ee38213729e119afe883b6&pid=1-s2.0-S0022123624003355-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142161642","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 : 2024-08-30DOI: 10.1016/j.jfa.2024.110644
Motivated by first-order conditions for extremal bodies of geometric functionals, we study a functional analytic notion of infinitesimal perturbations of convex bodies and give a full characterization of the set of realizable perturbations if the perturbed body is a polytope. As an application, we derive a necessary condition for polytopal maximizers of the isotropic constant.
{"title":"Small perturbations of polytopes","authors":"","doi":"10.1016/j.jfa.2024.110644","DOIUrl":"10.1016/j.jfa.2024.110644","url":null,"abstract":"<div><p>Motivated by first-order conditions for extremal bodies of geometric functionals, we study a functional analytic notion of infinitesimal perturbations of convex bodies and give a full characterization of the set of realizable perturbations if the perturbed body is a polytope. As an application, we derive a necessary condition for polytopal maximizers of the isotropic constant.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S002212362400332X/pdfft?md5=e462643d71502f41002932b5ab067bea&pid=1-s2.0-S002212362400332X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151201","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 : 2024-08-30DOI: 10.1016/j.jfa.2024.110652
The Kirchberg Embedding Problem (KEP) asks if every -algebra embeds into an ultrapower of the Cuntz algebra . Motivated by the recent refutation of the Connes Embedding Problem, we establish two computability-theoretic consequences of a positive solution to KEP. Both of our results follow from the a priori weaker assumption that there exists a locally universal -algebra with a computable presentation.
{"title":"Locally universal C⁎-algebras with computable presentations","authors":"","doi":"10.1016/j.jfa.2024.110652","DOIUrl":"10.1016/j.jfa.2024.110652","url":null,"abstract":"<div><p>The Kirchberg Embedding Problem (KEP) asks if every <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra embeds into an ultrapower of the Cuntz algebra <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Motivated by the recent refutation of the Connes Embedding Problem, we establish two computability-theoretic consequences of a positive solution to KEP. Both of our results follow from the a priori weaker assumption that there exists a locally universal <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra with a computable presentation.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142172920","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 : 2024-08-30DOI: 10.1016/j.jfa.2024.110659
We study the Kolmogorov law for a random sequence with prescribed radii so that it generates a Carleson measure almost surely, both for the Hardy space on the polydisc and the Hardy space on the unit ball, thus providing improved versions of previous results of the first two authors and of a separate result of Massaneda. In the polydisc, the geometry of such sequences is not well understood, so we proceed by studying the random Gramians generated by random sequences, using tools from the theory of random matrices. Another result we prove, and that is of its own relevance, is the law for a random sequence to be partitioned into M separated sequences with respect to the pseudo-hyperbolic distance, which is used also to describe the random sequences that are interpolating for the Bloch space on the unit disc almost surely.
我们研究了具有规定半径的随机序列的柯尔莫哥洛夫 0-1 定律,从而使其几乎肯定地产生卡列松度量,这既适用于多圆盘上的哈代空间,也适用于单位球上的哈代空间,从而提供了前两位作者先前结果和马萨内达单独结果的改进版本。在多圆盘上,人们对这类序列的几何结构还不太了解,因此我们利用随机矩阵理论的工具,研究随机序列产生的随机格拉米安。我们证明的另一个结果本身也具有相关性,那就是关于伪双曲距离的随机序列被分割成 M 个分离序列的 0-1 规律,该规律也用于描述几乎肯定插值为单位圆盘上布洛赫空间的随机序列。
{"title":"Random Carleson sequences for the Hardy space on the polydisc and the unit ball","authors":"","doi":"10.1016/j.jfa.2024.110659","DOIUrl":"10.1016/j.jfa.2024.110659","url":null,"abstract":"<div><p>We study the Kolmogorov <span><math><mn>0</mn><mo>−</mo><mn>1</mn></math></span> law for a random sequence with prescribed radii so that it generates a Carleson measure almost surely, both for the Hardy space on the polydisc and the Hardy space on the unit ball, thus providing improved versions of previous results of the first two authors and of a separate result of Massaneda. In the polydisc, the geometry of such sequences is not well understood, so we proceed by studying the random Gramians generated by random sequences, using tools from the theory of random matrices. Another result we prove, and that is of its own relevance, is the <span><math><mn>0</mn><mo>−</mo><mn>1</mn></math></span> law for a random sequence to be partitioned into <em>M</em> separated sequences with respect to the pseudo-hyperbolic distance, which is used also to describe the random sequences that are interpolating for the Bloch space on the unit disc almost surely.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003471/pdfft?md5=002a3ca282e7fe0e69899ce4b770adcb&pid=1-s2.0-S0022123624003471-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142161637","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 : 2024-08-28DOI: 10.1016/j.jfa.2024.110642
We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in with no ellipticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic p-harmonic functions or p-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of p-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.
我们在对各向异性(它只是 RN 中的一个规范,没有椭圆性或光滑性的要求,以便包括晶体情况)和初始数据作非常温和的假设下,构建了各向异性反向平均曲率流(A-IMCF)的弱解。通过莫泽引入的近似程序,我们的解是各向异性 p 谐函数或 p 容函数(变量改变后)的极限,我们得到了近似解(即 p 容函数的唯一性)和极限解的唯一性。我们的弱解概念仍然恢复了类似于 Huisken-Ilmanen 引入的变分和几何定义,但需要在更广泛的 BV 函数背景下工作。尽管如此,我们仍然获得了经典结果,如周长的连续性和指数增长,以及子级集的向外最小化特性。此外,通过假设内部滚动球条件(其中滑动的 Wulff 形状扮演球的角色)给出的额外规则性,我们证明了解的连续性并满足哈纳克不等式。最后,还建立了显式解的实例。
{"title":"Weak solutions of anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials","authors":"","doi":"10.1016/j.jfa.2024.110642","DOIUrl":"10.1016/j.jfa.2024.110642","url":null,"abstract":"<div><p>We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> with no ellipticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic <em>p</em>-harmonic functions or <em>p</em>-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of <em>p</em>-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of <em>BV</em>-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003306/pdfft?md5=b8aed4f15cb7f0e5872276ff50c4a2cd&pid=1-s2.0-S0022123624003306-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130089","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 : 2024-08-13DOI: 10.1016/j.jfa.2024.110625
We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to -actions on a compact space can be lifted to an isomorphism of their foliation -algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the -algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.
{"title":"A geometric Elliott invariant and noncommutative rigidity of mapping tori","authors":"","doi":"10.1016/j.jfa.2024.110625","DOIUrl":"10.1016/j.jfa.2024.110625","url":null,"abstract":"<div><p>We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-actions on a compact space can be lifted to an isomorphism of their foliation <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003136/pdfft?md5=a1cbbfe5bdbc4ef488f7c160f8a48b02&pid=1-s2.0-S0022123624003136-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076533","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}