Weighted Big Lipschitz algebras of analytic functions and closed ideals

Lipschitz interpolative nonlinear ideal procedure

We treat the general theory of nonlinear ideals and extend as many notions as possible from the linear theory to the nonlinear theory. We define nonlinear ideals with special properties which associate new non-linear ideals to given ones and establish several properties and characterizations of them. Building upon the results of U. Matter we define a Lipschitz interpolative nonlinear ideal procedure between metric spaces and Banach spaces and establish this class of Lipschitz operators is an injective Banach nonlinear ideal and show several standard basic properties for such class. Extending t…

The Lipschitz bounded approximation property for operator ideals

In this article, we introduce the Lipschitz bounded approximation property for operator ideals. With this notion, we extend the original work done by Godefroy and Kalton and give some partial answers on the equivalence between the bounded approximation property and the Lipschitz bounded approximation property based on an arbitrary operator ideal. Furthermore, we investigate the three space problems on the preceding bounded approximation properties.

On the action of Lipschitz functions on vector-valued random sums

Let $X$ be a Banach space and let $(ξ_j)_{j\ge 1}$ be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions are equivalent: (1). There exists a constant $K$ such that $$ \Bigl(\E\Big\|\sum_{j=1}^n ξ_j f(x_j)\Big\|^2\Bigr)^{\frac12} \leq K \n f\n_{\rm Lip} \Bigl(\E\Big\|\sum_{j=1}^n ξ_j x_j\Big\|^2\Bigr)^{\frac12} $$ for all Lipschitz functions $f:X\to X$ satisfying $f(0)=0$ and all finite sequences $x_1,...,x_n$ in $X$. (2). $X$ is isomorphic to a Hilbert space.

The approximation property and Lipschitz mappings on Banach spaces

We present an overview to the approximation property, paying especial attention to the recent results relating the approximation property to ideals of linear operators and Lipschitz ideals. We complete the paper with some new results on approximation of Lipschitz mappings and their relation to linear operator ideals.

Lipschitz spaces and M-ideals

For a metric space $(K,d)$ the Banach space $\Lip(K)$ consists of all scalar-valued bounded Lipschitz functions on $K$ with the norm $\|f\|_{L}=\max(\|f\|_{\infty},L(f))$, where $L(f)$ is the Lipschitz constant of $f$. The closed subspace $\lip(K)$ of $\Lip(K)$ contains all elements of $\Lip(K)$ satisfying the $\lip$-condition $\lim_{0<d(x,y)\to 0}|f(x)-f(y)|/d(x,y)=0$. For $K=([0,1],| {\cdot} |^α)$, $0<α<1$, we prove that $\lip(K)$ is a proper $M$-ideal in a certain subspace of $\Lip(K)$ containing a copy of $\ell^{\infty}$.

关于 Lipschitz 算子理想 Lip₀∘𝒜∘Lip₀

On the Lipschitz operator ideal $\text{Lip}_{0}\circ \mathcal A\circ \text{Lip}_{0}$

本文研究由 Banach 线性算子理想 𝒜 构造 Lipschitz 算子理想 Lip₀∘𝒜∘Lip₀，并用超积刻画其极大包络及线性算子成员关系。

Nonlinear Operator Ideals Between Metric Spaces and Banach Spaces (PART I)

In this paper we present part I of nonlinear operator ideals theory between metric spaces and Banach spaces. Building upon the definition of operator ideal between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear versions of operator ideals. We introduce several examples of nonlinear ideals and the relationships between them. For every space ideal $\mathsf{A}$ can be generated by a special nonlinear ideal which consists of those Lipschitz operators admitting a factorization through a Banach space $\mathbf{M}\in\mathsf{A}$. We investigate products and quotients of nonlinea…

On Lipschitz and d.c. surfaces of finite codimension in a Banach space

Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space, and properties of generated $σ$-ideals are studied. These $σ$-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.

Lipschitz稳定序列类：Lipschitz函数的Rademacher型与余型的一种方法

Lipschitz stable sequence classes: an approach to Rademacher type and cotype of Lipschitz functions

本文通过引入Lipschitz稳定序列类，将序列类方法从线性与多重线性理论扩展到Lipschitz设定，定义了$(Z,Y)$-求和Lipschitz函数空间，并证明其在稳定序列类下构成Banach Lipschitz理想，进而刻画了Lipschitz函数的Rademacher型与余型。

Closed ideals in some algebras of analytic functions

We obtain a complete description of closed ideals of the algebra $\mathcal{D}\cap \mathrm{lip}_α},$ $0<α\leq{1/2},$ where $\mathcal{D}$ is the Dirichlet space and $\mathrm{lip}_α}$ is the algebra of analytic functions satisfying the Lipschitz condition of order $α.$

Lipschitz estimates in quasi-Banach Schatten ideals

We study the class of functions $f$ on $\mathbb{R}$ satisfying a Lipschitz estimate in the Schatten ideal $\mathcal{L}_p$ for $0 < p \leq 1$. The corresponding problem with $p\geq 1$ has been extensively studied, but the quasi-Banach range $0 < p < 1$ is by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class $\dot{B}^{\frac{1}{p}}_{\frac{p}{1-p},p}(\mathbb{R})$ obey the estimate $$ \|f(A)-f(B)\|_{p} \leq C_{p}(\|f'\|_{L_{\infty}(\mathbb{R})}+\|f\|_{\dot{B}^{\frac{1}{p}}_{\frac{p}{1-p},p}(\mathbb{R})})\…

Dichotomy theorems for random matrices and closed ideals of operators on $\big(\bigoplus_{n=1}^\infty\ell_1^n \big)_{\mathrm{c}_0}$

We prove two dichotomy theorems about sequences of operators into $L_1$ given by random matrices. In the second theorem we assume that the entries of each random matrix form a sequence of independent, symmetric random variables. Then the corresponding sequence of operators either uniformly factor the identity operators on $\ell_1^k$ $(k\in\mathbb N$) or uniformly approximately factor through $\mathrm{c}_0$. The first theorem has a slightly weaker conclusion still related to factorization properties but makes no assumption on the random matrices. Indeed, it applies to operators defined on an ar…

关于 Dunkl 背景下的 Lipschitz 空间——半群方法

On Lipschitz spaces in the Dunkl setting -- semigroup approach

本文利用半群方法定义了 Dunkl 背景下的非齐次 Lipschitz 空间，并证明该空间与经典 Lipschitz 空间等价。

关于Banach空间单位球上A-紧全纯Lipschitz映射

A-compact holomorphic Lipschitz mappings on the unit ball of a Banach space

本文研究Banach空间单位球上A-紧全纯Lipschitz映射的结构与性质，探讨其作为Banach全纯Lipschitz理想的关系。

线性算子的Lipschitz延拓

Lipschitz extensions of linear operators

本文证明线性算子可延拓至极大Banach算子理想当且仅当其可延拓至相关Lipschitz算子理想，并推论特殊Lipschitz分解蕴含线性分解。

薛定谔框架下BMO与Lipschitz空间的若干刻画

Some characterizations of BMO and Lipschitz spaces in the Schrödinger setting

本文针对薛定谔算子相关的BMO和Lipschitz空间，给出若干新刻画，将已有研究结果推广到权类范围大于经典Muckenhoupt类的加权情形。

Schur bounded patterns和非闭合子主要化

Schur bounded patterns, submajorisation and operator Lipschitz functions

本文研究了非闭合子主要化的算子理想Schur bounded patterns，分析了Schatten理想C_p（0<p<1）的特性及非闭合子主要化的理想特征。

中心化 Takagi–van der Waerden 函数及其 Lipschitz 导数

Centered Takagi--van der Waerden functions and their Lipschitz derivatives

本文通过修改广义 Takagi–van der Waerden 函数，证明在无孤立点的度量空间中，任意闭子集上都存在连续函数使其各类 Lipschitz 导数恰在该集合上为无穷，并推广到封闭空间情形。

Banach 函数空间上的格 Lipschitz 叠加算子

Lattice Lipschitz superposition operators on Banach function spaces

本文刻画了 Banach 函数空间之间的格 Lipschitz 叠加算子，证明其良定义性等价于可由属于 Bochner 空间 $\mathcal M(X,Y)(μ, Lip_0(\mathbb R))$ 的强可测函数表示，并揭示了其与赋范张量积等映射类的关系。

Lipschitz映射与单调映射延拓的连续性

Continuity of extensions of Lipschitz maps and of monotone maps

本文研究 Hilbert 空间子集上 Lipschitz 映射与单调映射延拓的连续性，给出保持一致距离的 1-Lipschitz 延拓存在性的充分必要条件，并将结果推广到无穷维空间。

Operator-Lipschitz functions in Schatten-von Neumann classes

This paper resolves a number of conjectures in the perturbation theory of linear operators. Namely, we prove that every Lipschitz function is operator Lipschitz in the Schatten-von Neumann ideals $S^α$, $1 < α< \infty$. The negative result for $S^α$, $α= 1, \infty$ was earlier established by Yu. Farforovskaya in 1972.

Classifying the closed ideals of bounded operators on two families of non-separable classical Banach spaces

We classify the closed ideals of bounded operators acting on the Banach spaces $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{c_0} \oplus c_0(Γ)$ and $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{\ell_1} \oplus \ell_1(Γ)$ for every uncountable cardinal $Γ$.

Cone unrectifiable sets and non-differentiability of Lipschitz functions

We provide sufficient conditions for a set $E\subset\mathbb{R}^n$ to be a non-universal differentiability set, i.e. to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of $\mathbb{R}^n$ given by Alberti, Csörnyei and Preiss, which eventually led to the result of Jones and Csörnyei that for every Lebesgue null set $E$ in $\mathbb{R}^n$ there is a Lipschitz map $f:\mathbb{R}^n\to\mathbb{R}^n$ not differentiable at any point …

球面上和高斯空间中的集中不等式最优常数

Optimal constants in concentration inequalities on the sphere and in the Gauss space

本文研究球面和高斯空间中函数偏离中位数和均值的集中不等式，给出最优常数的界，改进了现有文献中的结果。

超度量空间上Lipschitz-free空间的对偶性

Duality of Lipschitz-free spaces over ultrametric spaces

本文刻画了超度量空间上Lipschitz-free空间成为对偶Banach空间的条件，证明其预对偶的M-嵌入性等价于度量空间正常，并否定Werner关于M-理想的问题，同时证明强极值点的存在性。

度量空间中的Takagi-van der Waerden函数及其Lipschitz导数

Takagi-van der Waerden functions in metric spaces and its Lipschitz derivatives

本文引入度量空间上的Takagi-van der Waerden函数，证明其大、小Lipschitz导数在非孤立点处为无穷大，并构造连续函数使Lipschitz导数恰在给定开集上发散。

Idéaux fermés de certaines algèbres de Beurling et application aux opérateurs à spectre dénombrable

We denote by $\bbt$ the unit circle and by $\bbd$ the unit disc of $\bbc$. Let $s$ be a non-negative real and $ω$ a weight such that $ω(n) = (1+n)^{s} \quad (n \geq 0)$ and such that the sequence $\dsp \Big(\frac{ω(-n)}{(1+n)^{s}} \Big)_{n \geq 0}$ is non-decreasing. We define the Banach algebra $$ A_ω(\bbt) = \Big\{f \in \calc(\bbt) : \big\| f \big\|_ω = \sum_{n = -\infty}^{+\infty} | \hat{f}(n) | ω(n) < +\infty \Big\}, $$ If $I$ is a closed ideal of $A_ω(\bbt)$, we set $h^{0}(I) = \Big\{z \in \bbt : f(z) = 0 \quad (f \in I) \Big\}$. We describe here all closed ideals $I$ of $A_ω(\bbt)$ such …

Lipschitz导数按半连续性与Baire极限函数的分类

Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions

本文引入广义半连续性，分类度量空间上Lipschitz导数，并证明赋范空间上小、大、局部Lipschitz导数与Baire极限函数的关系。

Singular traces and perturbation formulae of higher order

Let $H, V$ be self-adjoint operators such that $V$ belongs to the weak trace class ideal. We prove higher order perturbation formula $$τ\big(f(H+V)-\sum_{j=0}^{n-1}\frac{1}{j!}\frac{d^j}{dt^j} f(H+tV)\big|_{t=0}\big)=\int_{\mathbb{R}} f^{(n)}(t)\,dm_n(t),$$ where $τ$ is a trace on the weak trace class ideal and $m_n$ is a finite measure that is not necessarily absolutely continuous. This result extends the first and second order perturbation formulas of Dykema and Shripka, who generalised the Krein and Koplienko trace formulas to the weak trace class ideal. We also establish the perturbation f…

关于曲线平坦 Lipschitz 函数及其线性化

On curve-flat Lipschitz functions and their linearizations

本文证明 Lipschitz 映射线性化中若干算子理想重合，即 Dunford-Pettis、Radon-Nikodým 性质与不固定 $L_1$ 副本等价，并研究了相应的曲线平坦性度量推广。

Lipschitz自由空间中一个恰当的闭子空间包含线性对偶空间

A Proper Closed Subspace of the Lipschitz Dual Containing the Linear Dual

本文研究了Lipschitz自由空间中一个恰当的闭子空间，证明其作为偶空间的性质，并展示了该空间作为Banach代数的结构优势。

Hölder-Lipschitz函数在profinite群上的Titchmarsh定理

Titchmarsh Theorems for Hölder-Lipschitz functions on profinite groups

本文将Titchmarsh关于Hölder-Lipschitz函数Fourier变换的经典定理推广至可度量化profinite群，证明其与Sobolev正则性的关系，并刻画了Dini-Lipschitz类。

Lipschitz functions of perturbed operators

We prove that if $f$ is a Lipschitz function on $\R$, $A$ and $B$ are self-adjoint operators such that ${\rm rank} (A-B)=1$, then $f(A)-f(B)$ belongs to the weak space $\boldsymbol{S}_{1,\be}$, i.e., $s_j(A-B)\le{\rm const} (1+j)^{-1}$. We deduce from this result that if $A-B$ belongs to the trace class $\boldsymbol{S}_1$ and $f$ is Lipschitz, then $f(A)-f(B)\in\boldsymbol{S}_Ω$, i.e., $\sum_{j=0}^ns_j(f(A)-f(B))\le\const\log(2+n)$. We also obtain more general results about the behavior of double operator integrals of the form $Q=\iint(f(x)-f(y))(x-y)^{-1}dE_1(x)TdE_2(y)$, where $E_1$ and $E_2…

The boundedness of the Hilbert transformation from one rearrangement invariant Banach space into another and applications

In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.

Lipschitz spaces adapted to Schrödinger operators and regularity properties

Consider the Schrödinger operator $\mathcal{L}=-Δ+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse Hölder condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B V(y)^qdy\right)^{1/q}\le \frac{C}{|B|}\int_B V(y)dy, \, \text{ for some }q>n/2. \end{equation*} We define $Λ^α_{\mathcal{L}},\, 0<α<2,$ the class of measurable functions such that $$ \|ρ(\cdot)^{-α}f(\cdot)\|_\infty<\infty \quad \, \, \text{and}\:\: \quad \sup_{|z|>0}\frac{\|f(\cdot+z)+f(\cdot-z)-2f(\cdot)\|_\infty}{|z|^α}<\infty, $$ where $ρ$ is the critical radius function associated…