Journal of Functional Analysis最新文献

{"title":"On the radicality property for spaces of symbols of bounded Volterra operators","authors":"Carme Cascante ,&nbsp;Joan Fàbrega ,&nbsp;Daniel Pascuas ,&nbsp;José Ángel Peláez","doi":"10.1016/j.jfa.2024.110658","DOIUrl":"10.1016/j.jfa.2024.110658","url":null,"abstract":"<div><p>In <span><span>[1]</span></span> it is shown that the Bloch space <span><math><mi>B</mi></math></span> in the unit disc has the following radicality property: if an analytic function <em>g</em> satisfies that <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>∈</mo><mi>B</mi></math></span>, then <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>∈</mo><mi>B</mi></math></span>, for all <span><math><mi>m</mi><mo>≤</mo><mi>n</mi></math></span>. Since <span><math><mi>B</mi></math></span> coincides with the space <span><math><mi>T</mi><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>)</mo></math></span> of analytic symbols <em>g</em> such that the Volterra-type operator <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>z</mi></mrow></msubsup><mi>f</mi><mo>(</mo><mi>ζ</mi><mo>)</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>ζ</mi><mo>)</mo><mspace></mspace><mi>d</mi><mi>ζ</mi></math></span> is bounded on the classical weighted Bergman space <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>, the radicality property was used to study the composition of paraproducts <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>g</mi></math></span> on <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>. Motivated by this fact, we prove that <span><math><mi>T</mi><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>)</mo></math></span> also has the radicality property, for any radial weight <em>ω</em>. Unlike the classical case, the lack of a precise description of <span><math><mi>T</mi><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>)</mo></math></span> for a general radial weight, induces us to prove the radicality property for <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span> from precise norm-operator results for compositions of analytic paraproducts.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S002212362400346X/pdfft?md5=432f79852fe43e5d19c78c51c3225a0e&pid=1-s2.0-S002212362400346X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142161636","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":"On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori","authors":"Roberto Feola ,&nbsp;Jessica Elisa Massetti","doi":"10.1016/j.jfa.2024.110648","DOIUrl":"10.1016/j.jfa.2024.110648","url":null,"abstract":"<div><p>We consider a completely resonant nonlinear Schrödinger equation on the <em>d</em>-dimensional torus, for any <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span>, with polynomial nonlinearity of any degree <span><math><mn>2</mn><mi>p</mi><mo>+</mo><mn>1</mn></math></span>, <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>, which is gauge and translation invariant. We study the behaviour of <em>high</em> Sobolev <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>-norms of solutions, <span><math><mi>s</mi><mo>≥</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>&gt;</mo><mi>d</mi><mo>/</mo><mn>2</mn><mo>+</mo><mn>2</mn></math></span>, whose initial datum <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> satisfies an appropriate smallness condition on its <em>low</em> <span><math><msup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norms respectively. We prove a polynomial upper bound on the possible growth of the Sobolev norm <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> over finite but long time scale that is exponential in the regularity parameter <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. As a byproduct we get stability of the low <span><math><msup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math></span>-norm over such time interval. A key ingredient in the proof is the introduction of a suitable “modified energy” that provides an a priori upper bound on the growth. This is obtained by combining para-differential techniques and suitable tame estimates.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003367/pdfft?md5=96a57ec86be28722eb2014aec51b0e62&pid=1-s2.0-S0022123624003367-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151203","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":"Riesz operators and Lp-boundary representations for hyperbolic groups","authors":"Adrien Boyer ,&nbsp;Jean-Martin Paoli","doi":"10.1016/j.jfa.2024.110650","DOIUrl":"10.1016/j.jfa.2024.110650","url":null,"abstract":"<div><p>We investigate <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-boundary representations of hyperbolic groups. We prove that such representations are irreducible if and only if the corresponding Riesz operators are injective.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204683","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":"Jointly stationary solutions of periodic Burgers flow","authors":"Alexander Dunlap ,&nbsp;Yu Gu","doi":"10.1016/j.jfa.2024.110656","DOIUrl":"10.1016/j.jfa.2024.110656","url":null,"abstract":"<div><p>For the one dimensional Burgers equation with a random and periodic forcing, it is well-known that there exists a family of invariant measures, each corresponding to a different average velocity. In this paper, we consider the coupled invariant measures and study how they change as the velocity parameter varies. We show that the derivative of the invariant measure with respect to the velocity parameter exists, and it can be interpreted as the steady state of a diffusion advected by the Burgers flow.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151205","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":"Bessel functions on GL(n), I","authors":"Jack Buttcane","doi":"10.1016/j.jfa.2024.110657","DOIUrl":"10.1016/j.jfa.2024.110657","url":null,"abstract":"<div><p>In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on <span><math><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on <span><math><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on <span><math><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of <span><math><mi>G</mi><mi>L</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> Bessel functions.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142169256","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":"Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework","authors":"Shengxin Li ,&nbsp;Feng Xie","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}

{"title":"The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates","authors":"Yihong Du,&nbsp;Wenjie Ni","doi":"10.1016/j.jfa.2024.110649","DOIUrl":"10.1016/j.jfa.2024.110649","url":null,"abstract":"&lt;div&gt;&lt;p&gt;This is the second part of a two-part series devoted to an in depth understanding of the dynamical behaviour of the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In Part 1 &lt;span&gt;&lt;span&gt;[19]&lt;/span&gt;&lt;/span&gt;, we have shown that the long-time dynamics of this problem is characterised by a spreading-vanishing dichotomy, and there exists a threshold condition on the diffusion kernel &lt;span&gt;&lt;math&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is determined by an associated semi-wave problem established in &lt;span&gt;&lt;span&gt;[15]&lt;/span&gt;&lt;/span&gt;. In Part 2 here, we obtain more precise description of the spreading profile by focusing on some natural classes of kernel functions, including those satisfying &lt;span&gt;&lt;math&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≫&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. Our results for such kernels reveal a striking difference of behaviour from the pattern exhibited in the one dimension case &lt;span&gt;&lt;span&gt;[18]&lt;/span&gt;&lt;/span&gt; when &lt;em&gt;β&lt;/em&gt; crosses the value &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. More precisely, (a) when &lt;span&gt;&lt;math&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, we show that for &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≫&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; if &lt;span&gt;&lt;math&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; if &lt;span&gt;&lt;math&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, which is of the same pattern as in dimension one, namely we recover the result in &lt;span&gt;&lt;span&gt;[18]&lt;/span&gt;&lt;/span&gt; by letting &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; in the above statements; (b) when &lt;span&gt;&lt;math&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, the front has a finite spreading speed &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; i","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":"142161638","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 Widom–Sobolev formula for discontinuous matrix-valued symbols","authors":"Leon Bollmann,&nbsp;Peter Müller","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}

{"title":"Porous medium type reaction-diffusion equation: Large time behaviors and regularity of free boundary","authors":"Qingyou He","doi":"10.1016/j.jfa.2024.110643","DOIUrl":"10.1016/j.jfa.2024.110643","url":null,"abstract":"&lt;div&gt;&lt;p&gt;We consider the Cauchy problem of the porous medium type reaction-diffusion equation&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;em&gt;g&lt;/em&gt; is the given monotonic decreasing function with the density critical threshold &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; satisfying &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. We prove that the pressure &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; tends to the pressure critical threshold &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; at the time decay rate &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. If the initial density &lt;span&gt;&lt;math&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is compactly supported, we justify that the support &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of the density &lt;em&gt;ρ&lt;/em&gt; expands exponentially in time. Furthermore, we show that there exists a time &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; such that the pressure &lt;em&gt;P&lt;/em&gt; is Lipschitz continuous for &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, which is the optimal (sharp) regularity of the pressure, and the free surface &lt;span&gt;&lt;math&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;","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}

{"title":"On the optimal rate for the convergence problem in mean field control","authors":"Samuel Daudin ,&nbsp;François Delarue ,&nbsp;Joe Jackson","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}