Pointwise Lineability in Sequence Spaces

测度论中函数网与不可数函数序列的线性性

Lineability on nets and uncountable sequences of functions in measure theory

针对函数网会破坏控制收敛定理、单调收敛定理等经典结论的问题，本文研究反例函数网族的代数普遍性，并考察不可数序列情形。

Lineability criteria, with applications

Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Fa…

Lineability and spaceability for the weak form of Peano's theorem and vector-valued sequence spaces

Two new applications of a technique for spaceability are given in this paper. For the first time this technique is used in the investigation of the algebraic genericity property of the weak form of Peano's theorem on the existence of solutions of the ODE $u'=f(u)$ on $c_0$. The space of all continuous vector fields $f$ on $c_0$ is proved to contain a closed $\mathfrak{c}$-dimensional subspace formed by fields $f$ for which -- except for the null field -- the weak form of Peano's theorem fails to be true. The second application generalizes known results on the existence of closed $\mathfrak{c}$…

$\ell_{\infty}$ 的某些子集中的稠密线性和空间性

Dense lineability and spaceability in certain subsets of $\ell_{\infty}$

本文研究了 $\ell_\infty$ 中具有给定聚点数量的子集的稠密线性和空间性，证明了具有可数或有限聚点序列集合的稠密线性性质，并区分了它们的空间性差异。

关于至多具有有限个零坐标的序列

On Sequences with at Most a Finite Number of Zero Coordinates

本文研究仅含有限个零坐标的序列集合，借助 S-可线性化、逐点可线性化和 (alpha, beta)-可空间化等概念，刻画其代数与拓扑结构，并证明含 \\(c_0\\) 的 \\(\ell_p\\) 闭子空间中相应集合的 (alpha, c)-可空间化等价于 alpha 有限。

On the size of certain subsets of invariant Banach sequence spaces

The essence of the notion of lineability and spaceability is to find linear structures in somewhat chaotic environments. The existing methods, in general, use \textit{ad hoc} arguments and few general techniques are known. Motivated by the search of general methods, in this paper we formally extend recent results of G.\ Botelho and V.V. Fávaro on invariant sequence spaces to a more general setting. Our main results show that some subsets of invariant sequence spaces contain, up to the null vector, a closed infinite-dimensional subspace.

序列类与算子理想中的几何空间性

Geometric spaceability in sequence classes and operator ideals

本文提出了标准序列类的概念，研究了序列空间和算子理想中的（α，c）-空间性，扩展了现有文献并改进了相关结果。

部分度量值函数序列的统计收敛性

Statistical convergence of sequences of partial metric valued functions

本文引入部分度量值函数序列的统计逐点收敛、统计一致收敛及等统计收敛等概念，探究各类收敛性间的关系，拓展了部分度量值函数序列收敛性的相关研究。

Norm optimization problem for linear operators in classical Banach spaces

The main result of the paper shows that, for 1<p and 1<=q, a linear operator T from l_p to l_q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p=1). For 1<p (and q different from p), as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T from l_p to l_q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p=q). We also investigate lineability of the sets of norm-attaining and non-norm attaini…

The joint modulus of variation of metric space valued functions and pointwise selection principles

Given $T\subset\mathbb{R}$ and a metric space $M$, we introduce a nondecreasing sequence of pseudometrics $\{ν_n\}$ on $M^T$ (the set of all functions from $T$ into $M$), called the \emph{joint modulus of variation}. We prove that if two sequences of functions $\{f_j\}$ and $\{g_j\}$ from $M^T$ are such that $\{f_j\}$ is pointwise precompact, $\{g_j\}$ is pointwise convergent, and the limit superior of $ν_n(f_j,g_j)$ as $j\to\infty$ is $o(n)$ as $n\to\infty$, then $\{f_j\}$ admits a pointwise convergent subsequence whose limit is a conditionally regulated function. We illustrate the sharpness …

从线性角度看待几乎处处收敛与逐点收敛

Almost uniform vs. pointwise convergence from a linear point of view

本文从线性结构角度比较不同收敛模式，证明了存在大量向量子空间和代数结构，这些结构包含逐点几乎处处收敛但不几乎均匀收敛的序列，以及几乎均匀收敛但不逐点几乎处处收敛的序列。

点wise乘法算子在非代数情形下的本质范数

Essential norms of pointwise multipliers in the non-algebraic setting

本文计算了在不同Köthe空间之间作用的乘法算子的本质范数和弱本质范数，探讨了Banach序列空间和单位圆上解析函数空间的相关应用。

On the Brezis-Lieb Lemma without pointwise convergence

Brezis-Lieb lemma is a refinement of Fatou lemma providing an evaluation of the gap between the integral for a sequence and the integral for its pointwise limit. This note studies the question if such gap can be evaluated when there is no a.e. convergence. In particular, it gives the same lower bound for the gap in L^p as the gap in the Brezis-Lieb lemma (including the case vector-valued functions) provided that p is greater or equal than 3 and the sequence converges both weakly and weakly in the sense of a duality map. It also shows that the statement is false if p<3. An application is given …

泛函分析中若干网族的代数通有性

Algebraic genericity of certain families of nets in Functional Analysis

本文从线性角度研究泛函分析中多类异常网的代数通有性，涵盖弱收敛无界网、不满足Banach-Steinhaus定理的网、弱稠密范数无界网及关联网发散的收敛级数，推广并改进了拓扑向量空间与赋范空间中的若干已有结果。

非单射映射集合的空间性

Spaceability of sets of non-injective maps

该研究推广了非单射线性算子集合的线性性结果，证明一般映射线性空间中非单射映射子集含无限维子空间，并给出其在算子空间、算子理想及非线性映射空间中的应用。

Uniform convergence for convexification of dominated pointwise convergent continuous functions

The Lebesgue dominated convergence theorem of the measure theory implies that the Riemann integral of a bounded sequence of continuous functions over the interval [ 0,1] pointwise converging to zero, also converges to zero. The validity of this result is independent of measure theory, on the other hand, this result together with only elementary functional analysis, can generate measure theory itself. The mentioned result was also known before the appearance of measure theory, but the original proof was very complicated. For this reason this result, when presented in teaching, is generally obta…

The approximate variation to pointwise selection principles

Let $T\subset\mathbb{R}$, $M$ be a metric space with metric $d$, and $M^T$ be the set of all functions mapping $T$ into $M$. Given $f\in M^T$, we study the properties of the approximate variation $\{V_\varepsilon(f)\}_{\varepsilon>0}$, where $V_\varepsilon(f)$ is the greatest lower bound of Jordan variations $V(g)$ of functions $g\in M^T$ such that $d(f(t),g(t))\le\varepsilon$ for all $t\in T$. The notion of $\varepsilon$-variation $V_\varepsilon(f)$ was introduced by Frańková [Math. Bohem. 116 (1991), 20-59] for intervals $T=[a,b]$ in $\mathbb{R}$ and $M=\mathbb{R}^N$ and extended to the gene…

Algebras of functions with Fourier coefficients in weighted Orlicz sequence spaces

We prove that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to $\ell^1\cap\ell^Φ_{ϕ,w}$ (resp. to $\ell^1\cap\ell^Ψ_{ψ,\varrho}$), where $\ell^Φ_{ϕ,w}$ and $\ell^Ψ_{ψ,\varrho}$ are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences \[ ϕ=\{ϕ_n\},\quad ψ=\{ψ_n\},\quad w=\{w_n\},\quad \varrho=\{\varrho_n\} \] increase and satisfy the $Δ_2$-condition.

Metastable convergence theorems

The dominated convergence theorem implies that if (f_n) is a sequence of functions on a probability space taking values in the interval [0,1], and (f_n) converges pointwise a.e., then the sequence of integrals converges to the integral of the pointwise limit. Tao has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence (f_n) and the underlying space. We prove a slight strengthening of Tao's theorem which, moreover…

Kadec norms on spaces of continuous functions

We study the existence of pointwise Kadec renormings for Banach spaces of the form $C(K)$. We show in particular that such a renorming exists when $K$ is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if $C(K_1)$ has a pointwise Kadec renorming and $K_2$ belongs to the class of spaces obtained by closing the class of compact metrizable spaces under inverse limits of transfinite continuous sequences of retractions, then $C(K_1\times K_2)$ has a pointwise Kadec renorming. We also prove a version of t…

Pointwise Convergence of Ergodic Averages in Orlicz Spaces

We show that for each Orlicz space properly contained in L^1 there is a sequence along which the ergodic averages converge for functions in the Orlicz space, but diverge for all f in L^1. This extends the work of K. Reinhold, who, building on the work of A. Bellow,constructed a sequence for which the averages converge a.e. for every f in L^p, p>q, but diverge for some f in L^q. Our method, introduced by Bellow and extended by Reinhold and M. Wierdl, is perturbation.

Convergence of Functions and their Moreau Envelopes on Hadamard Spaces

A well known result of H. Attouch states that the Mosco convergence of a sequence of proper convex lower semicontinuous functions defined on a Hilbert space is equivalent to the pointwise convergence of the associated Moreau envelopes. In the present paper we generalize this result to Hadamard spaces. More precisely, while it has already been known that the Mosco convergence of a sequence of convex lower semicontinuous functions on a Hadamard space implies the pointwise convergence of the corresponding Moreau envelopes, the converse implication was an open question. We now fill this gap. Our r…

A new proof of James' sup theorem

We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson (1977) : "If a normed space $E$ does not contain any asymptotically isometric copy of $\ell^1(\IN)$, then every bounded sequence of $E'$ has a normalized block sequence pointwise converging to 0".

局部紧群上沿Reiter序列的逐点遍历定理

Pointwise Ergodic Theorems Along Reiter Sequences on Locally Compact Groups

本文针对容许Reiter序列的局部紧群，证明沿该序列的遍历平均的极大不等式，建立其双边几乎一致收敛性，并推导出非交换Wiener-Wintner型定理。

Divergence of wavelet series: A multifractal analysis

We show the relevance of a multifractal-type analysis for pointwise convergence and divergence properties of wavelet series: Depending on the sequence space which the wavelet coefficients sequence belongs to, we obtain deterministic upper bounds for the Hausdorff dimensions of the sets of points where a given rate of divergence occurs, and we show that these bounds are generically optimal, according to several notions of genericity.

Maps of several variables of finite total variation and Helly-type selection principles

Given a map from a rectangle in the n-dimensional real Euclidean space into a metric semigroup, we introduce a concept of the total variation, which generalizes a similar concept due to T. H. Hildebrandt (1963) for real functions of two variables and A. S. Leonov (1998) for real functions of n variables, and study its properties. We show that the total variation has many classical properties of Jordan's variation such as the additivity, generalized triangle inequality and sequential lower semicontinuity. We prove two variants of a pointwise selection principle of Helly-type, one of which is as…

A note on the weak* and pointwise convergence of BV functions

We study pointwise convergence properties of weakly* converging sequences $\{u_i\}_{i \in {\mathbb N}}$ in $\mathrm{BV}({\mathbb R}^n)$. We show that, after passage to a suitable subsequence (not relabeled), we have pointwise convergence $u_i^*(x)\to u^*(x)$ of the precise representatives for all $x\in {\mathbb R}^n \setminus E$, where the exceptional set $E \subset {\mathbb R}^n$ has on the one hand Hausdorff dimension at most $n-1$, and is on the other hand also negligible with respect to the Cantor part of $|D u|$. Furthermore, we discuss the optimality of these results.

一类加权Orlicz-Sobolev序列空间嵌入的紧性

On the compactness of embeddings for a class of weighted Orlicz-Sobolev sequence spaces

本文引入一类加权Orlicz-Sobolev序列空间，证明其连续性与紧性准则，连续性源于Orlicz函数逐点比较，紧性结合Schauder基与局部条件，并给出实例与开放问题。

Pointwise convergence of partial functions: The Gerlits-Nagy Problem

For a set $X\sbst\R$, let $B(X)\sbst\R^X$ denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product $\R^X$. Assume that for each countable $A\sbst B(X)$, each $f$ in the closure of $A$ is in the closure of $A$ under pointwise limits of sequences of partial functions. We show that in this case, $B(X)$ is countably Fréchet--Urysohn, that is, each point in the closure of a countable set is a limit of a sequence of elements of that set. This solves a problem of Arnold Miller. The continuous version of this problem is equivalent to a notorious op…

bv_p 空间中的非线性复合算子：连续性与紧性

Nonlinear composition operators in bv_p spaces: continuity and compactness

本文研究 bv_p 等序列空间上非线性复合算子的逐点、一致、局部一致、Lipschitz 与 Hölder 连续性，并给出其紧性刻画。

从$C_p(X)$到$c_0$中对称序列理想的连续线性满射

Continuous Linear Surjections from $C_p(X)$ onto Symmetric Sequence Ideals in $c_0$

该研究针对$C_p(X)$空间的商空间相关公开问题，证明了若$C_p(X)$到带逐点拓扑的$c_0$对称序列理想有连续线性满射，则该理想必为$c_0$本身，结合已有结果给出了满射存在的完整刻画。

Weak type $(1, 1)$ estimates for maximal functions along $1$-regular sequences of integers

We show the pointwise convergence of the averages \[ \mathcal{A}_N f(x) = \frac{1}{\# \mathbf{B}_N} \sum_{n \in \mathbf{B}_N} f(x + n) \] for $f \in \ell^1(\mathbb{Z})$ where $\mathbf{B}_N = \mathbf{B} \cap [1, N]$, and $\mathbf{B}$ is a $1$-regular sequence of integers, for example $\mathbf{B} = \{\lfloor n \log n \rfloor : n \in \mathbb{N}\}$.

The approximate variation of univariate uniform space valued functions and pointwise selection principles

Let $T\subset\mathbb{R}$ and $(X,\mathcal{U})$ be a uniform space with an at most countable gage of pseudometrics $\{d_p:p\in\mathcal{P}\}$ of the uniformity $\mathcal{U}$. Given $f\in X^T$ (=the family of all functions from $T$ into $X$), the approximate variation of $f$ is the two-parameter family $\{V_{\varepsilon,p}(f):\varepsilon>0,p\in\mathcal{P}\}$, where $V_{\varepsilon,p}(f)$ is the greatest lower bound of Jordan's variations $V_p(g)$ on $T$ with respect to $d_p$ of all functions $g\in X^T$ such that $d_p(f(t),g(t))\le\varepsilon$ for all $t\in T$. We establish the following pointwise…

Pointwise multipliers in Hardy-Orlicz spaces, and interpolation

We study multipliers of Hardy-Orlicz spaces $\mH_Φ$ which are strictly contained between $\bigcup_{p>0}H^p$ and so-called ``big'' Hardy-Orlicz spaces. Big Hardy-Orlicz spaces, carrying an algebraic structure, are equal to their multiplier algebra, whereas in classical Hardy spaces $H^p$, the multipliers reduce to $H^{\infty}$. For Hardy-Orlicz spaces $\mH_Φ$ between these two extremal situations and subject to some conditions, we exhibit multipliers that are in Hardy-Orlicz spaces the defining functions of which are related to $Φ$. Even if the results do not entirely characterize the multiplie…

Orientation-preserving Young measures

We prove a characterization result in the spirit of the Kinderlehrer-Pedregal Theorem for Young measures generated by gradients of Sobolev maps satisfying the orientation-preserving constraint, that is the pointwise Jacobian is positive almost everywhere. The argument to construct the appropriate generating sequences from such Young measures is based on a variant of convex integration in conjunction with an explicit lamination construction in matrix space. Our generating sequence is bounded in $L^p$ for $p$ less than the space dimension, a regime in which the pointwise Jacobian loses some of i…

点态函数空间可分商空间的一致失败

Separable quotients of pointwise function spaces and relative consistency

在Jensen菱形原则下构造紧致零维空间K，使C_p(K)无无限维Hausdorff可分线性商，并证明可分商与可度量商存在性等价。

Order and uo-convergence in spaces of continuous functions

We present several characterizations of uo-convergent nets or sequences in spaces of continuous functions $C(Ω)$, $C_b(Ω)$, $C_0(Ω)$, and $C^\infty(Ω)$, extending results of [vdW18]. In particular, it is shown that a sequence uo-converges iff it converges pointwise on a co-meagre set. We also characterize order bounded sets in spaces of continuous functions. This leads to characterizations of order convergence.

局部紧空间上半群作用的逐点连续遍历性

Continuous pointwise ergodicity for semigroup actions on locally compact spaces

该研究针对局部紧空间上的半群作用，建立了逐点连续遍历性的算子理论刻画，推广了紧空间群作用的经典结果，且适用于无Følner序列的半群，还给出了相关动力学实例。

Functions of Baire class one

Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index $β$ and the convergence index $γ$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $β(f) \leq ω^{ξ_1} \cdot ω^{ξ_2}$ for some countable ordinals $ξ_1$ and $ξ_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise su…

特殊 null 序列的空性

Spaceability of special families of null sequences of holomorphic functions

本文研究了特殊 null 序列的空性，证明了两个闭合无限维子空间的存在，分别涉及点集收敛与紧致收敛的性质。

Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces

In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of $RCD^*(K,N)$-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in $RCD^*(K,N)$-spaces. We use then these results to initiate the study of Weyl's law in the $RCD$ setting