Journal of Functional Analysis最新文献

{"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}

{"title":"Wave front sets of nilpotent Lie group representations","authors":"Julia Budde,&nbsp;Tobias Weich","doi":"10.1016/j.jfa.2024.110684","DOIUrl":"10.1016/j.jfa.2024.110684","url":null,"abstract":"<div><div>Let <em>G</em> be a nilpotent, connected, simply connected Lie group with Lie algebra <span><math><mi>g</mi></math></span>, and <em>π</em> a unitary representation of <em>G</em>. In this article we prove that the wave front set of <em>π</em> coincides with the asymptotic cone of the orbital support of <em>π</em>, i.e. <span><math><mrow><mi>WF</mi></mrow><mo>(</mo><mi>π</mi><mo>)</mo><mo>=</mo><mrow><mi>AC</mi></mrow><mo>(</mo><msub><mrow><mo>⋃</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mrow><mi>supp</mi></mrow><mo>(</mo><mi>π</mi><mo>)</mo></mrow></msub><msub><mrow><mi>O</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo>⊂</mo><mi>i</mi><msup><mrow><mi>g</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is the coadjoint Kirillov orbit associated to the irreducible unitary representation <span><math><mi>σ</mi><mo>∈</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003720/pdfft?md5=83be36def1c70aa00f9f837a0f297c17&pid=1-s2.0-S0022123624003720-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312835","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":"Absolute continuity of degenerate elliptic measure","authors":"Mingming Cao ,&nbsp;Kôzô Yabuta","doi":"10.1016/j.jfa.2024.110673","DOIUrl":"10.1016/j.jfa.2024.110673","url":null,"abstract":"<div><div>Let <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> be an open set whose boundary may be composed of pieces of different dimensions. Assume that Ω satisfies the quantitative openness and connectedness, and there exist doubling measures <em>m</em> on Ω and <em>μ</em> on ∂Ω with appropriate size conditions. Let <span><math><mi>L</mi><mi>u</mi><mo>=</mo><mo>−</mo><mi>div</mi><mo>(</mo><mi>A</mi><mi>∇</mi><mi>u</mi><mo>)</mo></math></span> be a real (not necessarily symmetric) degenerate elliptic operator in Ω. Write <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>L</mi></mrow></msub></math></span> for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>(</mo><mi>μ</mi><mo>)</mo></math></span>, (ii) the Dirichlet problem for <em>L</em> is solvable in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> for some <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, (iii) every bounded null solution of <em>L</em> satisfies Carleson measure estimates with respect to <em>μ</em>, (iv) the conical square function is controlled by the non-tangential maximal function in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> for all <span><math><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> for any null solution of <em>L</em>, and (v) the Dirichlet problem for <em>L</em> is solvable in <span><math><mi>BMO</mi><mo>(</mo><mi>μ</mi><mo>)</mo></math></span>. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>L</mi></mrow></msub></math></span> with respect to <em>μ</em> in terms of local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> estimates of the truncated conical square function for any bounded null solution of <em>L</em>. This is also equivalent to the finiteness <em>μ</em>-almost everywhere of the truncated conical square function for any bounded null solution of <em>L</em>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003616/pdfft?md5=768991f70c40bd283f96e3c4b9cba196&pid=1-s2.0-S0022123624003616-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312831","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":"Decay estimates for Beam equations with potential in dimension three","authors":"Miao Chen ,&nbsp;Ping Li ,&nbsp;Avy Soffer ,&nbsp;Xiaohua Yao","doi":"10.1016/j.jfa.2024.110671","DOIUrl":"10.1016/j.jfa.2024.110671","url":null,"abstract":"<div><p>This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><mo>(</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>V</mi><mo>)</mo><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span></span></span> in dimension three, where <em>V</em> is a real-valued and decaying potential. Assume that zero is a regular point of <span><math><mi>H</mi><mo>=</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>V</mi></math></span>, we first prove the following optimal time decay estimates of the solution operators<span><span><span><math><msub><mrow><mo>‖</mo><mi>cos</mi><mo>⁡</mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>≲</mo><mo>|</mo><mi>t</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mspace></mspace><mspace></mspace><mtext>and</mtext><msub><mrow><mo>‖</mo><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></mrow><mrow><msqrt><mrow><mi>H</mi></mrow></msqrt></mrow></mfrac><msub><mrow><mi>P</mi></mrow><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>≲</mo><mo>|</mo><mi>t</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>.</mo></math></span></span></span> Moreover, if zero is a resonance of <em>H</em>, then time decay of the solution operators also is considered. It is noted that a first-kind resonance does not affect the decay rates of the propagator operators <span><math><mi>cos</mi><mo>⁡</mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></math></span> and <span><math><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></mrow><mrow><msqrt><mrow><mi>H</mi></mrow></msqrt></mrow></mfrac></math></span>, but their decay will be significantly changed for the second and third-kind resonances.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142273960","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":"Equivalence of block sequences in Schreier spaces and their duals","authors":"R.M. Causey,&nbsp;A. Pelczar-Barwacz","doi":"10.1016/j.jfa.2024.110674","DOIUrl":"10.1016/j.jfa.2024.110674","url":null,"abstract":"<div><p>We prove that any normalized block sequence in a Schreier space <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>ξ</mi></mrow></msub></math></span>, of arbitrary order <span><math><mi>ξ</mi><mo>&lt;</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, admits a subsequence equivalent to a subsequence of the canonical basis of some Schreier space. The analogous result is proved for dual spaces to Schreier spaces. Using these results, we examine the structure of strictly singular operators on Schreier spaces and show that there are <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>c</mi></mrow></msup></math></span> many closed operator ideals on a Schreier space of any order, its dual and bidual space.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274602","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}