本文证明，对于$p>0$，$L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$是$(\alpha,\mathfrak{c})$-空间性的当且仅当$\alpha<\aleph_{0}$。该问题最早出现在[V. Fávaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020) 27-46]中，作者得到了$p>0$时$L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$的$(1,\mathfrak{c})$-空间性。此后其他学者继续寻求该问题的最终答案，并取得了一些部分结果。由于序列空间已有类似结论，这一结果的真实性在预料之中。
英文摘要：
In this note we prove that, for $p>0$, $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ is $(α,\mathfrak{c})$-spaceable if, and only if, $α<\aleph_{0}$. Such a problem first appears in [V. Fávaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020) 27-46], where the authors get the $(1,\mathfrak{c})$-spaceability of $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ for $p>0$. The definitive answer to this problem continued to be sought by other authors, and some partial answers were obtained. The veracity of this result was expected, as a similar result is known for sequence spaces.
Spaceability and algebrability of sets of nowhere integrable functions
证明了$[0,1]$中处处本质有界的Lebesgue可积函数集是空间可积的，改进了[F. J. García-Pacheco，M. Martín和J. B. Seoane-Sepúlveda. \textit{Lineability, spaceability, and algebrability of certain subsets of function spaces,}台湾数学杂志，\textbf{13}（2009），no.4，1257- 1269]，并且它是强$\mathfrak{c}$ -代数的。证明了具有稠密跳跃间断集的有界变差函数集的强$\mathfrak{c}$ -代数性和不可分空间性。作为推论，给出了应用于Lebesgue-nowhere-Riemann可积函数和Riemann-nowhere-Newton可积函数的集合。此外，我们证明了一组Kurzweil可积的功能，这不是Lebesgue可积的是空间可（在Alexievicz规范），但不是1-代数。我们还证明了存在无穷维可微函数向量空间$S$，使得$S$的$C([0,1])$ -闭包的每个元素都是Kurzweil可积函数的本原，与[V.I. Gurariy，\textit{Subspaces and bases in spaces of continuous functions (Russian),} Dokl…
In this paper, we study spaceability of subsets of generalized Orlicz and Lebesgue spaces associated to Banach function space. Also, we give some sufficient conditions for spaceability of subsets of a general Banach space which improves an important result on this topic. As an application, it is shown that the set of all bounded linear operators which are not positive semidefinite on a separable Hilbert space is spaceable.
We present the example of l^p spaces, where we examine results of topological and algebraic genericity and spaceability. At the end of the paper we include a project with other chains of spaces, mainly of holomorphic functions, as Hardy spaces on the disc.
We investigate lineability/spaceability problems within the setting of multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contemporary geometric notions of pointwise-lineability and $(α,β)$-lineability. Among other results, we prove that the class of multilinear operators taking values on $\ell_q \, (0<q \leq \infty)$ that are absolutely but not multiple summing is pointwise $\mathfrak{c}$-spaceable when non-empty. Spaceability in other classes, such as Dunford-Pettis operators and multilinear operators on general summable scalar families, is also stud…
We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.
Albiac and Wojtaszczyk introduced property (A) to characterize $1$-greedy bases. Later, Dilworth et al. generalized the concept to $C$-property (A), where the case $C = 1$ gives property (A). They (among other results) characterized greedy bases by unconditionality and $C$-property (A). In this paper, we extend the definition of the so-called A-property constant to (A,$τ$)-property constants and use the extension to obtain new estimates for various Lebesgue parameters. Furthermore, we study the relation among (A,$τ$)-property constants and other well-known constants when $τ$ varies.
For $μ>β>0$, the generalized Stieltjes operators $$ \mathcal{S}_{β,μ} f(t):={t^{μ-β}}\int_0^\infty {s^{β-1}\over (s+t)^μ}f(s)ds, \qquad t>0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(α)}(t^α)$ (where $α\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < β- \pp < μ$, then operators $\mathcal{S}_{β,μ}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Cesáro operator on $\mathcal{T}_p^{(α)}(t^α)$ . We calculate and represent explicitly their spectrum…
Generalized vectorial Lebesgue and Bochner integration theory
本文发展了可和函数的Lebesgue空间和Bochner空间理论。它是H. Lebesgue，S.巴拿赫，S. Bochner，G. Fubini，S. Saks，F. Riesz，N. Dunford，P. Halmos，和其他贡献者这个理论。该理论的构造是基于任何抽象空间X中集合的prering上的测度的概念。理论的发展不需要空间X的拓扑结构。Prering上的测度推广了抽象勒贝格测度的概念。这些测量是现成的，并且没有必要为了理论的发展而预先将它们扩展到西格玛环上。理论发展的基本工具是可和函数的Lebesgue-Bochner空间的构造和表征，如Bogdanowicz的论文“A Generalization of the Lebesgue-Bochner-Stieltjes Integral and a New Approach to the Theory of Integration”，Proc. of Nat. Acad. Sci. USA，Vol.53，No.3，（1965），p.492 - 498
Embeddability of $\ell_p$-spaces into mixed-norm Lebesgue spaces in connection with the validity of vector-valued extensions of the Riesz--Fischer Theorem
In this paper, we consider a new weak norm, iterated weak norm in Lebesgue spaces with mixed norms. We study properties of the mixed weak norm and the iterated weak norm and present the relationship between the two weak norms. Even for the ordinary Lebesgue spaces, the two weak norms are not equivalent and any one of them can not control the other one. We give some convergence and completeness results for both weak norms. We study the convergence in truncated norm, which is a substitution of the convergence in measure for mixed Lebesgue spaces. And we give a characterization of the convergence…
We give a fundament for Berezin's analytic $Ψ$do considered in \cite{Berezin71} in terms of Bargmann images of Pilipovi{ć} spaces. We deduce basic continuity results for such $Ψ$do, especially when the operator kernels are in suitable mixed weighted Lebesgue spaces and act on certain weighted Lebesgue spaces of entire functions. In particular, we show how these results imply well-known continuity results for real $Ψ$do with symbols in modulation spaces, when acting on other modulation spaces.
We study the classical problem of identifying the structure of $P^2(μ)$, the closure of analytic polynomials in the Lebesgue space $L^2(μ)$ of a compactly supported Borel measure $μ$ living in the complex plane. In his influential work, Thomson showed that the space decomposes into a full $L^2$-space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures $μ$ supported on the closed unit disk $\overline{\mathbb{D}}$ which have a part on the open disk $\mathbb{D}$ which is similar to the Lebesgue area measure, and a part on the unit …
On Convolution in Variable Lebesgue Spaces and Applications to Fractional Navier_Stokes Equations
Department of Computer Science, Faculty of Science and Technology, University of Ahmed Zabana（阿哈迈德·扎巴纳大学科学技术学院计算机科学系）；Laboratory of Functional Analysis and Geometry of Spaces, University of Mohamed Boudiaf（穆罕默德·布迪亚夫大学函数分析与空间几何实验室）
Operators of Harmonic Analysis in Weighted Spaces with Non-standard Growth
近年来，人们对非标准增长的函数空间，即变指数Lebesgue空间的研究日益增多。对于齐性空间上的加权空间，我们给出了Rubio de Francia外推定理的一个变形。应用这个外推定理，得到了变指数加权Lebesgue空间中各种调和分析算子，如极大算子、奇异算子、位势算子、Fourier乘子、三角傅里叶级数部分和的控制算子等在这类空间中的有界性.也给出了它们的向量值类似物。
It is proved the existence of large algebraic structures \break --including large vector subspaces or infinitely generated free algebras-- inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.
Let $X$ be a ball Banach function space on ${\mathbb R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on the associate space of the convexification of $X$, the authors prove that, for any locally integrable function $f$ with $\|\,|\nabla f|\,\|_{X}<\infty$, $$\sup_{λ\in(0,\infty)}λ\left \|\left|\left\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >λ|\cdot-y|^{\frac{n}{q}+1}\right\}\right|^{\frac{1}{q}} \right\|_X\sim \|\,|\nabla f|\,\|_X$$ with the positive equivalence constants independent of $f$, where the index $q\in(0,\i…
Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove that $S$ is meager if and only if there exists an ordinary limit point of $(x_n)$ which is not a statistical cluster point of $(x_n)$. This provides a non-analogue between measure and category.
