Journal of Functional Analysis最新文献

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Continuous dependence on initial data for the solutions of 3-D anisotropic Navier-Stokes equations 三维各向异性纳维-斯托克斯方程解对初始数据的连续依赖性

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-17 DOI: 10.1016/j.jfa.2024.110689

In this paper, we prove the continuous dependence on the initial data for the solutions of 3-D incompressible anisotropic Navier-Stokes equations in the functional space

X_{s} (T) \overset{def}{=} {u : u \in C ([0, T]; H^{0, s}) with \nabla_{h} u \in L^{2} (] 0, T [; H^{0, s})}

for

s > \frac{1}{2}

. We also show the non-uniform continuity of the data-to-solution map in

C ([0, T]; H^{0, s})

for

s > \frac{1}{2}

, which makes sharp contrast with the corresponding result for the classical 3-D Navier-Stokes equations.

本文证明了在 s>12 时，函数空间 Xs(T)=def{u:u∈C([0,T];H0,s)with∇hu∈L2(]0,T[;H0,s)} 中三维不可压缩各向异性纳维-斯托克斯方程解对初始数据的连续依赖性。我们还展示了 s>12 时 C([0,T];H0,s) 中数据到解图的非均匀连续性，这与经典三维纳维-斯托克斯方程的相应结果形成了鲜明对比。

引用次数: 0

Douglas-Rudin approximation theorem for operator-valued functions on the unit ball of Cd Cd 单位球上算子值函数的道格拉斯-鲁丁近似定理

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-16 DOI: 10.1016/j.jfa.2024.110685

Douglas and Rudin proved that any unimodular function on the unit circle

T

can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of

C^{d}

. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result (2009) [4] on the approximation of matrix-valued unimodular functions on

T

道格拉斯和鲁丁证明，单位圆 T 上的任何单调函数都可以通过内函数的商均匀逼近。我们将这一结果推广到定义在 Cd 的开放单位球边界上的算子值单模函数。我们的证明技术结合了单元算子的谱定理和标量情况下的道格拉斯-鲁丁定理，将结果引导到算子值情况。这就产生了一个新的证明，也是对 Barclay 关于 T 上矩阵值单模函数逼近的结果（2009 年）[4] 的重要推广。

引用次数: 0

Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data 具有奇异势和平方可积分数据的非光滑薛定谔方程的可解性

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-16 DOI: 10.1016/j.jfa.2024.110680

We develop a holomorphic functional calculus for first-order operators DB to solve boundary value problems for Schrödinger equations

- div A \nabla u + a V u = 0

in the upper half-space

R_{+}^{n + 1}

with

n \in N

. This relies on quadratic estimates for DB, which are proved for coefficients

A, a, V

that are independent of the transversal direction to the boundary, and comprised of a complex-elliptic pair

(A, a)

that are bounded and measurable, and a singular potential V in either

L^{n / 2} (R^{n})

or the reverse Hölder class

B^{q} (R^{n})

with

q \geq \max {\frac{n}{2}, 2}

. In the latter case, square function bounds are also shown to be equivalent to non-tangential maximal function bounds. This allows us to prove that the (Dirichlet) Regularity and Neumann boundary value problems with

L^{2} (R^{n})

-data are well-posed if and only if certain boundary trace operators defined by the functional calculus are isomorphisms. We prove this property when the principal coefficient matrix A has either a Hermitian or block structure. More generally, the set of all complex coefficients for which the boundary value problems are well-posed is shown to be open.

我们开发了一阶算子 DB 的全形函数微积分，以解决上半空间 R+n+1 中 n∈N 的薛定谔方程 -divA∇u+aVu=0 的边界值问题。这依赖于对 DB 的二次估计，其系数 A,a,V 与边界横向方向无关，由有界可测的复椭圆对 (A,a) 和 Ln/2(Rn)或反向荷尔德类 Bq(Rn)（q≥max{n2,2}）中的奇异势 V 组成。在后一种情况下，平方函数边界也被证明等价于非切线最大函数边界。这使我们能够证明，当且仅当某些由函数微积分定义的边界迹算子是同构的时候，具有 L2(Rn)-data 的（狄利克特）正则性和诺伊曼边界值问题是好求的。当主系数矩阵 A 具有赫米特结构或块结构时，我们将证明这一性质。更广义地说，边界值问题得到很好解决的所有复系数集合是开放的。

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引用次数: 0

Classification of equivariantly O2-stable amenable actions on nuclear C⁎-algebras 核 C⁎-代数上等变 O2 稳定可配作用的分类

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-16 DOI: 10.1016/j.jfa.2024.110683

Given a second-countable, locally compact group G, we consider amenable G-actions on separable, stable, nuclear

C^{⁎}

-algebras that are isometrically shift-absorbing and tensorially absorb the trivial action on the Cuntz algebra

O_{2}

. We show that such actions are classified up to cocycle conjugacy by the induced G-action on the primitive ideal space. In the special case when G is exact, we prove a unital version of our classification theorem. For compact groups, we obtain a classification up to conjugacy.

给定一个二次可数局部紧密群 G，我们考虑可分离、稳定、核 C⁎-原子团上的可处理 G 作用，这些作用等效地吸收移位，并张量地吸收 Cuntz 代数 O2 上的琐细作用。我们的研究表明，这种作用是由原始理想空间上的诱导 G 作用分类到共轭循环的。在 G 是精确的特殊情况下，我们证明了我们的分类定理的单元版本。对于紧凑群，我们得到了直至共轭的分类。

引用次数: 0

High moments of the SHE in the clustering regimes 聚类机制中的 SHE 高矩阵

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-16 DOI: 10.1016/j.jfa.2024.110675

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 [75] to prove an n-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.

我们分析了随机热方程（SHE）的高矩，将其转换为有吸引力的布朗粒子（BPs），即通过成对吸引力漂移相互作用的布朗运动。在粒子趋于聚集的缩放状态下，我们证明了吸引力布朗粒子经验度量的大偏差原理（LDP）。在 delta（-like）初始条件下，我们描述了速率函数的唯一最小值，并将该最小值与 Kardar-Parisi-Zhang (KPZ) 方程在上尾部的时空极限形状联系起来。本文的结果被用在同行论文[75]中，证明了 KPZ 方程的 n 点上尾 LDP，并描述了相应的时空极限形状。

引用次数: 0

Singular extension of critical Sobolev mappings under an exponential weak-type estimate 指数弱型估计下临界索波列夫映射的奇异扩展

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-16 DOI: 10.1016/j.jfa.2024.110681

Given

m \in N ∖ {0}

and a compact Riemannian manifold

N

, we construct for every map u in the critical Sobolev space

W^{m / (m + 1), m + 1} (S^{m}, N)

, a map

U : B_{1}^{m + 1} \to N

whose trace is u 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.

给定 m∈N∖{0} 和一个紧凑的黎曼流形 N，我们为临界索波列夫空间 Wm/(m+1),m+1(Sm,N)中的每一个映射 u 构建一个映射 U:B1m+1→N，其迹线为 u 并且满足指数弱型索波列夫估计。该结果及其证明可延伸至边界超平面上映射的半空间，以及边界球面上映射的无穷远处的双曲空间。

引用次数: 0

Higher index theory for spaces with an FCE-by-FCE structure 具有逐FCE结构的空间的高指数理论

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-16 DOI: 10.1016/j.jfa.2024.110679

Let ${(1 \to N_{n} \to G_{n} \to Q_{n} \to 1)}_{n \in N}$ be a sequence of extensions of finite groups. Assume that the coarse disjoint unions of ${(N_{n})}_{n \in N}$ , ${(G_{n})}_{n \in N}$ and ${(Q_{n})}_{n \in N}$ have bounded geometry. The sequence ${(G_{n})}_{n \in N}$ is said to have an FCE-by-FCE structure, if the sequence ${(N_{n})}_{n \in N}$ and the sequence ${(Q_{n})}_{n \in N}$ admit a fibred coarse embedding into Hilbert space. In this paper, we prove the coarse Novikov conjecture holds for the sequence ${(G_{n})}_{n \in N}$ with an FCE-by-FCE structure.

设（1→Nn→Gn→Qn→1）n∈N 是有限群的扩展序列。假设 (Nn)n∈N、(Gn)n∈N 和 (Qn)n∈N 的粗糙不相接的联合具有有界几何。如果序列(Nn)n∈N 和序列(Qn)n∈N 允许纤维粗嵌入到希尔伯特空间，则称序列(Gn)n∈N 具有 FCE-by-FCE 结构。本文证明了具有 FCE-by-FCE 结构的序列 (Gn)n∈N 的粗诺维科夫猜想成立。

引用次数: 0

Density of compactly supported smooth functions CC∞(Rd) in Musielak-Orlicz-Sobolev spaces W1,Φ(Ω) Musielak-Orlicz-Sobolev 空间 W1,Φ(Ω) 中紧凑支撑的光滑函数 CC∞(Rd)的密度

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-16 DOI: 10.1016/j.jfa.2024.110677

We investigate here the density of the set of the restrictions from

C_{C}^{\infty} (R^{d})

C_{C}^{\infty} (Ω)

in the Musielak-Orlicz-Sobolev space

W^{1, Φ} (Ω)

. It is a continuation of article [15], where we have studied density of

C_{C}^{\infty} (R^{d})

W^{k, Φ} (R^{d})

for

k \in N

. The main theorem states that for an open subset

Ω \subset R^{d}

with its boundary of class

C^{1}

, and Musielak-Orlicz function Φ satisfying condition (A1) which is a sort of log-Hölder continuity and the growth condition

Δ_{2}

, the set of restrictions of functions from

C_{C}^{\infty} (R^{d})

to Ω is dense in

W^{1, Φ} (Ω)

. We obtain a corresponding result in variable exponent Sobolev space

W^{1, p (\cdot)} (Ω)

under the assumption that the exponent

p (x)

is essentially bounded on Ω and

Φ (x, t) = t^{p (x)}

t \geq 0

x \in Ω

, satisfies the log-Hölder condition.

我们在此研究在 Musielak-Orlicz-Sobolev 空间 W1,Φ(Ω) 中从 CC∞(Rd)到 CC∞(Ω)的限制集的密度。这是文章[15]的继续，我们在文章[15]中研究了 k∈N 时 Wk,Φ(Rd) 中 CC∞(Rd)的密度。主定理指出，对于边界为 C1 类的开放子集 Ω⊂Rd，以及满足 log-Hölder 连续性条件 (A1) 和增长条件 Δ2 的 Musielak-Orlicz 函数 Φ，从 CC∞(Rd)到 Ω 的函数限制集在 W1,Φ(Ω) 中是密集的。假设指数 p(x) 在 Ω 上基本上是有界的且Φ(x,t)=tp(x), t≥0, x∈Ω, 满足 log-Hölder 条件，我们将在变指数 Sobolev 空间 W1,p(⋅)(Ω) 中得到相应的结果。

引用次数: 0

An integrable bound for rough stochastic partial differential equations with applications to invariant manifolds and stability 粗糙随机偏微分方程的可积分约束及其在不变流形和稳定性方面的应用

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-16 DOI: 10.1016/j.jfa.2024.110676

We study semilinear rough stochastic partial differential equations as introduced in Gerasimovičs and Hairer (2019) [31]. We provide $L^{p} (Ω)$ -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.

我们研究的是 Gerasimovičs 和 Hairer (2019) [31] 中引入的半线性粗糙随机偏微分方程。我们为方程由合适的高斯过程驱动时的解及其线性化提供了 Lp(Ω)-integrable 先验边界。利用巴拿赫空间的乘法遍历定理，我们可以推导出线性化方程在静止点附近存在李亚普诺夫谱。我们还提供了静止点周围存在的局部稳定流形、不稳定流形和中心流形。在所有 Lyapunov 指数都为负的情况下，可以推导出局部指数稳定性。我们用几个例子来说明我们的发现。

引用次数: 0

A probabilistic approach to Lorentz balls ℓq,1n 洛伦兹球 ℓq,1n 的概率方法

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-09-14 DOI: 10.1016/j.jfa.2024.110682

We develop a probabilistic approach to study the volumetric and geometric properties of unit balls $B_{q, 1}^{n}$ of finite-dimensional Lorentz sequence spaces $ℓ_{q, 1}^{n}$ . More precisely, we show that the empirical distribution of a random vector $X^{(n)}$ 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 $k \in N$ of coordinates of $X^{(n)}$ as $n \to \infty$ . Moreover, we prove a central limit theorem for the largest coordinate of $X^{(n)}$ , demonstrating a quite different behavior than in the case of the $ℓ_{q}^{n}$ 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 $ℓ_{q, 1}^{n}$ and $ℓ_{p}^{n}$ balls.

我们开发了一种概率方法来研究有限维洛伦兹序列空间 ℓq,1n 的单位球 Bq,1n 的体积和几何特性。更确切地说，我们证明了均匀分布在其体积归一化单位球上的随机向量 X(n) 的经验分布弱收敛于具有明确给定密度的紧凑支撑对称概率分布；因此，我们得到了对于 X(n) 坐标的任意固定数 k∈N 的弱 Poincaré-Maxwell-Borel 原则，即 n→∞。此外，我们还证明了 X(n) 最大坐标的中心极限定理，证明了与ℓqn 球截然不同的行为，在ℓqn 球的极限中出现了冈贝尔分布。最后，我们证明了关于体积归一化 ℓq,1n 和 ℓpn 球交点的渐近体积的谢赫特曼-施穆克恩施拉格（Schechtman-Schmuckenschläger）式结果。

引用次数: 0

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