arXiv 1709.082622017-09math.FA

Properties of a Hilbertian Norm for Perimeter

Felipe Hernandez

英文摘要：

A recent paper of Jerison and Figalli proved a relationship between the $H^{1/2}$ norms of smoothed out indicator functions of sets and their perimeter. We continue this line of investigation and extend it in two ways. First, we describe a description of the situation with general functions of bounded variation, and show that a related quantity controls the size of the jump set. Second, we provide an exact formula in the case of a set of finite perimeter. Several questions remain and are presented here.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

半希尔伯特空间算子的新半范数及其应用

New semi-norm of semi-Hilbertian space operators and its application

本研究提出半希尔伯特空间算子的一种新半范数，推广了A-数值半径与A-算子半范数，推导其基本性质与上下界，并借此得到更精确的A-数值半径上界。

2021-07-20math.FA

半希尔伯特空间中A-有界算子的A-近似点谱

$A$-approximate point spectrum of $A$-bounded operators in semi-Hilbertian spaces

该研究聚焦复半希尔伯特空间上的算子谱理论，刻画A-有界算子的A-近似点谱，分析A^(1/2)-伴随算子张量积的对应谱性质，并推导A-正规算子的相关性质，丰富了半希尔伯特空间算子理论的内容。

2024-03-08math.FA

半希尔伯特空间中算子和的$A$-范数与$A$-数值半径的不等式及其应用

Inequalities for the $A$-Norm and $A$-Numerical Radius of Operator Sums in Semi-Hilbertian Spaces with Applications

本文对半希尔伯特空间中算子和的$A$-范数与$A$-数值半径建立新不等式，改进广义三角不等式并给出算子乘积与交换子的新边界，可应用于量子力学和偏微分方程，证明了不等式的最优性与优势。

2025-07-08math.FAmath.OA

BMO-type norms related to the perimeter of sets

In this paper we consider an isotropic variant of the $BMO$-type norm recently introduced by Bourgain, Brezis and Mironescu. We prove that, when considering characteristic functions of sets, this norm is related to the perimeter. A byproduct of our analysis is a new characterization of the perimeter of sets in terms of this norm, independent of the theory of distributions.

2014-07-08math.FA

Wasserstein重心凸性检测Hilbert几何

Wasserstein Barycenter Convexity Detects Hilbertian Geometry

School of Mathematics, Shandong University（山东大学数学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）

证明Boltzmann熵在Wasserstein重心处的凸性足以区分Hilbert空间与一般Banach空间，从而提供内蕴的最优传输测试。

2026-06-29math.FAmath.MG

On Schatten restricted norms

We consider norms on a complex separable Hilbert space such that $\langle aξ,ξ\rangle\leq\|ξ\|^2\leq\langle bξ,ξ\rangle$ for positive invertible operators $a$ and $b$ that differ by an operator in the Schatten class. We prove that these norms have unitarizable isometry groups, our proof uses a generalization of a fixed point theorem for isometric actions on positive invertible operators. As a result, if the isometry group does not leave any finite dimensional subspace invariant, then the norm must be Hilbertian. That is, if a Hilbertian norm is changed to a close non-Hilbertian norm, then the …

2020-02-20math.FAmath.MG

更多相关论文收起更多论文

希尔伯特空间中的 Kahane--Salem--Zygmund 不等式：极值函数与定量间隙

Hilbertian Kahane--Salem--Zygmund Inequalities: Extremizers and Quantitative Gaps

Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Universidade Federal do Maranhão（马拉尼昂联邦大学）

该研究针对有限维希尔伯特空间上系数为±1的实多重线性型，确定了K_{r,n}=1的条件，推导了等式不成立时的范数间隙，还给出了渐近结果、四阶矩估计及相关构造。

2026-08-11math.FA

半希尔伯特空间中算子的$A$-数值半径的若干上界及其应用

Certain Upper Bounds on the $A$-Numerical Radius of Operators in Semi-Hilbertian Spaces and Their Applications

本文在半希尔伯特空间框架下研究算子的$A$-数值半径，通过建立新不等式改进现有上界、细化三角不等式，并结合实例验证有效性，为数值半径不等式及算子理论应用提供更精确的理论支撑。

2025-07-07math.FAmath.OA

Minimizers of anisotropic perimeters with cylindrical norms

We study various regularity properties of minimizers of the $Φ$--perimeter, where $Φ$ is a norm. Under suitable assumptions on $Φ$ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

2016-04-01math.APmath.FA

Complex Interpolation between Hilbert, Banach and Operator spaces

Motivated by a question of Vincent Lafforgue, we study the Banach spaces $X$ satisfying the following property: there is a function $\vp\to Δ_X(\vp)$ tending to zero with $\vp>0$ such that every operator $T\colon L_2\to L_2$ with $\|T\|\le \vp$ that is simultaneously contractive (i.e. of norm $\le 1$) on $L_1$ and on $L_\infty$ must be of norm $\le Δ_X(\vp)$ on $L_2(X)$. We show that $Δ_X(\vp)\in O(\vp^α)$ for some $α>0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $θ$-Hilbertian spaces for some $ θ>0$ (see Corollary \ref{comcor4.3}), where $θ$-Hilbertian is meant in…

2008-12-17math.FAmath.OA

半 Hilbert 空间中 $A$-正交性保持与 Blanco-Koldobsky-Turnšek 定理

On $A$-orthogonality preservation and Blanco-Koldobsky-Turnšek theorem in semi-Hilbert spaces

本文研究半 Hilbert 空间中 $A$-有界算子对点态 $A$-正交性的保持，给出完整刻画，并据此描述范数达到集及判定 $A$-等距。

2025-07-25math.FA

Fractional Sobolev norms and BV functions on manifolds

The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on compact manifolds. As an application the special case of sets of finite perimeter is considered.

2018-06-07math.APmath.FAmath.MG

From Kernel Methods to Neural Networks: A Unifying Variational Formulation

École polytechnique fédérale de Lausanne (EPFL)（洛桑联邦理工学院）

The minimization of a data-fidelity term and an additive regularization functional gives rise to a powerful framework for supervised learning. In this paper, we present a unifying regularization functional that depends on an operator and on a generic Radon-domain norm. We establish the existence of a minimizer and give the parametric form of the solution(s) under very mild assumptions. When the norm is Hilbertian, the proposed formulation yields a solution that involves radial-basis functions and is compatible with the classical methods of machine learning. By contrast, for the total-variation…

2022-06-29cs.LGmath.FAmath.OC