Journal of Functional Analysis最新文献

英文中文

Quantitative observability for one-dimensional Schrödinger equations with potentials 带电势的一维薛定谔方程的定量可观测性

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-19 DOI: 10.1016/j.jfa.2024.110695

Pei Su , Chenmin Sun , Xu Yuan

In this note, we prove the quantitative observability with an explicit control cost for the 1D Schrödinger equation over

R

with real-valued, bounded continuous potential on thick sets. Our proof relies on different techniques for low-frequency and high-frequency estimates. In particular, we extend the large time observability result for the 1D free Schrödinger equation in Theorem 1.1 of Huang-Wang-Wang [20] to any short time. As another byproduct, we extend the spectral inequality of Lebeau-Moyano [27] for real-analytic potentials to bounded continuous potentials in the one-dimensional case.

在本论文中，我们证明了 R 上的一维薛定谔方程的定量可观测性，该方程在厚集上具有实值、有界的连续势。我们的证明依赖于不同的低频和高频估计技术。特别是，我们将 Huang-Wang-Wang [20] 的定理 1.1 中的一维自由薛定谔方程的大时间可观测性结果扩展到任何短时间。作为另一个副产品，我们将 Lebeau-Moyano [27] 针对实解析势的谱不等式扩展到一维情况下的有界连续势。

引用次数: 0

On the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi equations 论超二次汉密尔顿-雅可比方程中霍尔德半矩的改进

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-19 DOI: 10.1016/j.jfa.2024.110692

Marco Cirant

We show in this paper that maximal

L^{q}

-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic γ-growth in the gradient holds in the full range

q > (N + 2) \frac{γ - 1}{γ}

. Our approach is based on new

\frac{γ - 2}{γ - 1}

-Hölder estimates, which are consequence of the decay at small scales of suitable nonlinear space and time Hölder quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.

我们在本文中证明，对于右边无约束且梯度超二次方γ增长的时变粘性汉密尔顿-雅可比方程，最大 Lq 不规则性在整个 q>(N+2)γ-1γ 范围内成立。我们的方法基于新的γ-2γ-1-霍尔德估计，这是合适的非线性空间和时间霍尔德商在小尺度上衰减的结果。这是通过证明合适的振荡估计而获得的，这些振荡估计还反过来给出了全解的一些利乌维尔式结果。

引用次数: 0

Approximation of SBV functions with possibly infinite jump set 可能具有无限跳跃集的 SBV 函数的近似值

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-19 DOI: 10.1016/j.jfa.2024.110686

Sergio Conti , Matteo Focardi , Flaviana Iurlano

We prove an approximation result for functions

u \in S B V (Ω; R^{m})

such that ∇u is p-integrable,

1 \leq p < \infty

, and

g_{0} (| [u] |)

is integrable over the jump set (whose

H^{n - 1}

measure is possibly infinite), for some continuous, nondecreasing, subadditive function

g_{0}

, with

g_{0}^{- 1} (0) = {0}

. The approximating functions

u_{j}

are piecewise affine with piecewise affine jump set; the convergence is that of

L^{1}

for

u_{j}

and the convergence in energy for

| \nabla u_{j} |^{p}

and

g ([u_{j}], ν_{u_{j}})

for suitable functions g. In particular,

u_{j}

converges to u BV-strictly, area-strictly, and strongly in BV after composition with a bilipschitz map. If in addition

H^{n - 1} (J_{u}) < \infty

, we also have convergence of

H^{n - 1} (J_{u_{j}})

H^{n - 1} (J_{u})

我们证明了函数 u∈SBV(Ω;Rm)的近似结果：对于某个连续的、非递减的、次正函数 g0，∇u 是 p 可积分的，1≤p<∞，且 g0(|[u]|) 在跳跃集（其 Hn-1 度量可能是无限的）上是可积分的，g0-1(0)={0}。近似函数 uj 是片断仿射的，具有片断仿射跳跃集；uj 的收敛性是 L1 的收敛性，对于合适的函数 g，|∇uj|p 和 g([uj],νuj) 的收敛性是能量的收敛性。此外，如果 Hn-1(Ju)<∞，我们也会得到 Hn-1(Juj)向 Hn-1(Ju) 收敛的结果。

引用次数: 0

Sobolev smoothing estimates for bilinear maximal operators with fractal dilation sets 具有分形扩张集的双线性最大算子的索波列夫平滑估计值

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-19 DOI: 10.1016/j.jfa.2024.110694

Tainara Borges , Benjamin Foster , Yumeng Ou

Given a hypersurface

S \subset R^{2 d}

, we study the bilinear averaging operator that averages a pair of functions over S, as well as more general bilinear multipliers of limited decay and various maximal analogs. Of particular interest are bilinear maximal operators associated to a fractal dilation set

E \subset [1, 2]

; in this case, the boundedness region of the maximal operator is associated to the geometry of the hypersurface and various notions of the dimension of the dilation set. In particular, we determine Sobolev smoothing estimates at the exponent

L^{2} \times L^{2} \to L^{2}

using Fourier-analytic methods, which allow us to deduce additional

L^{p}

improving bounds for the operators and sparse bounds and their weighted corollaries for the associated multi-scale maximal functions. We also extend the method to study analogues of these questions for the triangle averaging operator and biparameter averaging operators. In addition, some necessary conditions for boundedness of these operators are obtained.

给定一个超曲面 S⊂R2d，我们研究将一对函数平均到 S 上的双线性平均算子，以及更一般的有限衰减双线性乘子和各种最大类似算子。尤其令人感兴趣的是与分形扩张集 E⊂[1,2] 相关的双线性最大算子；在这种情况下，最大算子的有界区域与超曲面的几何形状和扩张集维度的各种概念相关。特别是，我们利用傅立叶分析方法确定了指数 L2×L2→L2 的索波列夫平滑估计，从而推导出了算子的额外 Lp 改进边界，以及相关多尺度最大函数的稀疏边界及其加权推论。我们还扩展了该方法，以研究三角平均算子和双参数平均算子的类似问题。此外，我们还得到了这些算子有界性的一些必要条件。

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引用次数: 0

Generalized hyperbolicity, stability and expansivity for operators on locally convex spaces 局部凸空间上算子的广义双曲性、稳定性和扩张性

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-18 DOI: 10.1016/j.jfa.2024.110696

Nilson C. Bernardes Jr. , Blas M. Caraballo , Udayan B. Darji , Vinícius V. Fávaro , Alfred Peris

We introduce and study the notions of (generalized) hyperbolicity, topological stability and (uniform) topological expansivity for operators on locally convex spaces. We prove that every generalized hyperbolic operator on a locally convex space has the finite shadowing property. Contrary to what happens in the Banach space setting, hyperbolic operators on Fréchet spaces may fail to have the shadowing property, but we find additional conditions that ensure the validity of the shadowing property. Assuming that the space is sequentially complete, we prove that generalized hyperbolicity implies the strict periodic shadowing property, but we also show that the hypothesis of sequential completeness is essential. We show that operators with the periodic shadowing property on topological vector spaces have other interesting dynamical behaviors, including the fact that the restriction of such an operator to its chain recurrent set is topologically mixing and Devaney chaotic. We prove that topologically stable operators on locally convex spaces have the finite shadowing property and the strict periodic shadowing property. As a consequence, topologically stable operators on Banach spaces have the shadowing property. Moreover, we prove that generalized hyperbolicity implies topological stability for operators on Banach spaces. We prove that uniformly topologically expansive operators on locally convex spaces are neither Li-Yorke chaotic nor topologically transitive. Finally, we characterize the notion of topological expansivity for weighted shifts on Fréchet sequence spaces. Several examples are provided.

我们引入并研究了局部凸空间上算子的（广义）双曲性、拓扑稳定性和（均匀）拓扑扩张性等概念。我们证明，局部凸空间上的每个广义双曲算子都具有有限阴影特性。与巴拿赫空间的情况相反，弗雷谢特空间上的双曲算子可能不具有阴影性质，但我们找到了确保阴影性质有效性的附加条件。假设空间是连续完备的，我们证明广义双曲性意味着严格的周期阴影性质，但我们也证明连续完备性假设是必不可少的。我们证明拓扑向量空间上具有周期阴影特性的算子还有其他有趣的动力学行为，包括这样一个算子对其链循环集的限制是拓扑混合和德瓦尼混沌的。我们证明局部凸空间上的拓扑稳定算子具有有限阴影特性和严格周期阴影特性。因此，巴拿赫空间上的拓扑稳定算子具有阴影性质。此外，我们证明广义双曲性意味着巴拿赫空间上算子的拓扑稳定性。我们证明了局部凸空间上均匀拓扑扩张算子既不是李-约克混沌的，也不是拓扑传递的。最后，我们描述了弗雷谢特序列空间上加权移动的拓扑扩张性概念。我们提供了几个例子。

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引用次数: 0

Statistical mechanics of the wave maps equation in dimension 1 + 1 1 + 1 维波图方程的统计力学

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-18 DOI: 10.1016/j.jfa.2024.110688

Zdzisław Brzeźniak , Jacek Jendrej

We study wave maps with values in

S^{d}

, defined on the future light cone

{| x | \leq t} \subset R^{1 + 1}

, with prescribed data at the boundary

{| x | = t}

. Based on the work of Keel and Tao, we prove that the problem is well-posed for locally absolutely continuous boundary data. We design a discrete version of the problem and prove that for every absolutely continuous boundary data, the sequence of solutions of the discretised problem converges to the corresponding continuous wave map as the mesh size tends to 0.

Next, we consider the boundary data given by the

S^{d}

-valued Brownian motion. We prove that the sequence of solutions of the discretised problems has a subsequence that converges in law in the topology of locally uniform convergence. We argue that the resulting random field can be interpreted as the wave-map evolution corresponding to the initial data given by the Gibbs distribution.

我们研究在未来光锥{|x|≤t}⊂R1+1上定义的具有 Sd 值的波图，边界 {|x|=t} 有规定数据。基于 Keel 和 Tao 的研究，我们证明了对于局部绝对连续的边界数据，该问题可以很好地求解。我们设计了该问题的离散版本，并证明对于每个绝对连续的边界数据，随着网格尺寸趋于 0，离散化问题的解序列会收敛到相应的连续波图。我们证明，离散化问题的解序列有一个在局部均匀收敛拓扑中规律收敛的子序列。我们认为，由此产生的随机场可以解释为与吉布斯分布给出的初始数据相对应的波图演化。

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引用次数: 0

Hypocoercivity in Hilbert spaces 希尔伯特空间中的下协迫性

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-18 DOI: 10.1016/j.jfa.2024.110691

F. Achleitner , A. Arnold , V. Mehrmann , E.A. Nigsch

The concept of hypocoercivity for linear evolution equations with dissipation is discussed and equivalent characterizations that were developed for the finite-dimensional case are extended to separable Hilbert spaces. Using the concept of a hypocoercivity index, quantitative estimates on the short-time and long-time decay behavior of a hypocoercive system are derived. As a useful tool for analyzing the structural properties, an infinite-dimensional staircase form is also derived and connections to linear systems and control theory are presented. Several examples illustrate the new concepts and the results are applied to the Lorentz kinetic equation.

本文讨论了具有耗散的线性演化方程的次协迫性概念，并将有限维情况下的等效特征扩展到可分离的希尔伯特空间。利用超矫顽力指数的概念，得出了对超矫顽力系统的短时和长时衰减行为的定量估计。作为分析结构特性的有用工具，还导出了无穷维楼梯形式，并介绍了与线性系统和控制理论的联系。几个例子说明了新概念，并将结果应用于洛伦兹动力学方程。

引用次数: 0

Wolff potential estimates and Wiener criterion for nonlocal equations with Orlicz growth 具有奥利兹增长的非局部方程的沃尔夫势估计和维纳准则

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-18 DOI: 10.1016/j.jfa.2024.110690

Minhyun Kim , Ki-Ahm Lee , Se-Chan Lee

We prove the Wolff potential estimates for nonlocal equations with Orlicz growth. As an application, we obtain the Wiener criterion in this framework, which provides a necessary and sufficient condition for boundary points to be regular. Our approach relies on the fine analysis of superharmonic functions in view of nonlocal nonlinear potential theory.

我们证明了具有奥立兹增长的非局部方程的沃尔夫势估计。作为应用，我们在此框架下获得了维纳准则，它为边界点的规则性提供了必要且充分的条件。我们的方法依赖于根据非局部非线性势理论对超谐函数进行精细分析。

引用次数: 0

The elliptic maximal function 椭圆最大函数

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-17 DOI: 10.1016/j.jfa.2024.110693

Juyoung Lee , Sanghyuk Lee , Sewook Oh

We study the elliptic maximal functions defined by averages over ellipses and rotated ellipses which are multi-parametric variants of the circular maximal function. We prove that those maximal functions are bounded on

L^{p}

for some

p \neq \infty

. For this purpose, we obtain some sharp multi-parameter local smoothing estimates.

我们研究了由椭圆和旋转椭圆的平均值定义的椭圆最大函数，它们是圆最大函数的多参数变体。我们证明，对于某些 p≠∞，这些最大函数在 Lp 上是有界的。为此，我们得到了一些尖锐的多参数局部平滑估计值。

引用次数: 0

Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I I 型非紧密李群的局部解析扭转和相对解析扭转

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-17 DOI: 10.1016/j.jfa.2024.110687

A. Della Vedova , M. Spreafico

Let G be a (non compact) connected, simply connected, locally compact, second countable Lie group, either abelian or unimodular of type I, and let ρ be an irreducible unitary representation of G. Then, we define the analytic torsion of G localised at the representation ρ. The idea of considering localised invariants is due to Brodzki, Niblo, Plymen and Wright [5], and was exploited in [31] to define a localised eta function. Next, let Γ be a discrete co compact subgroup of G. We use the localised analytic torsion to define the relative analytic torsion of the pair

(G, Γ)

, and we prove that the last coincides with the Lott

L^{2}

analytic torsion of a covering space. We illustrate these constructions analysing in some details two examples: the abelian case, and the case

G = H

, the Heisenberg group.

让 G 是一个（非紧凑）连通的、简单连通的、局部紧凑的、第二可数李群，是 I 型的非等边或单模态，让 ρ 是 G 的一个不可还原的单元表示。接下来，让 Γ 成为 G 的离散协紧凑子群。我们使用局部化解析扭转来定义一对 (G,Γ) 的相对解析扭转，并证明最后一个解析扭转与覆盖空间的 Lott L2 解析扭转重合。我们以两个例子详细分析了这些构造：非等边情况和海森堡群 G=H 的情况。

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