Sergey V. Astashkin

• Professor
• Chair at Samara National Research University Russian Federation

We prove that every measurable function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\,[0,a]\rightarrow \mathbb {C}$$\end{document} such that \documentclass[12pt]{...

We investigate connections between upper/lower estimates for Banach lattices and the notion of relative s-decomposability, which has roots in interpolation theory. To get a characterization of relatively s-decomposable Banach lattices in terms of the above estimates, we assign to each Banach lattice X two sequence spaces \(X_{U}\) and \(X_{L}\) tha...

We study absolute summability of inclusions of r.i. function spaces. It appears that such properties are closely related, or even determined by absolute summability of inclusions of subspaces spanned by the Rademacher system in respective r.i. spaces. Our main result states that for $1<p<2$ the inclusion $X_p\subset L^p$ is $(q,1)$-absolutely summi...

We prove that both multiple Rademacher system and Rademacher chaos possess the property of random unconditional convergence in the space $L_\infty$. This fact combined with some intimate connections between $L_\infty$-norms of linear combinations of elements of these systems and some special norms of matrices of their coefficients allows us to esta...

In terms of variations, a sufficient condition for the uniform convergence of sequences of continuous functions is proved. Using this result, we obtain an addition to the classical Helly theorem on the selection of convergent sequences of functions with uniformly bounded variations in the case when the limit function is continuous. Also, by using a...

Let X be a separable rearrangement invariant space on \((0,\infty )\). If the intersection \((X \cap L_{\infty })(0,\infty )\) contains a complemented subspace isomorphic to \({\ell }_2\), then X contains a complemented sublattice lattice-isomorphic to \({\ell }_2\). Moreover, we prove that the space \((X+L_{\infty })(0,\infty )\) cannot be isomorp...

The main aim of the survey is to present results of the last decade on the description of subspaces spanned by independent functions in $L_p$-spaces and Orlicz spaces on the one hand, and in general rearrangement invariant spaces on the other. A new approach is proposed, which is based on a combination of results in the theory of rearrangement inva...

We study the subspaces of the Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,…$, of a function $f\in L_M$, $\int_0^1 f(t) dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and condi...

Изучаются подпространства пространств Орлича $L_M$, порожденные независимыми копиями $f_k$, $k=1,2,…$, функций $f\in L_M$, $\int_0^1 f(t) dt=0$. В терминах растяжений функции $f$ получено описание сильно вложенных подпространств этого типа, а также найдены условия, гарантирующие, что нормы функций единичного шара в таком подпространстве равностепен...

We study subspaces of Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,\dots$, of a function $f\in L_M$, $\int_0^1 f(t)\,dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditio...

We prove that every measurable function $f:\,[0,a]\to\mathbb{C}$ such that $|f|=1$ a.e. on $[0,a]$ is an extreme point of the unit ball of the Lorentz space $\Lambda(\varphi)$ on $[0,a]$ whenever $\varphi$ is a not linear, strictly increasing, concave, continuous function on $[0,a]$ with $\varphi(0)=0$. As a consequence, we complement the classical...

The closed linear span of the Rademacher functions in $L^2[0,1]$ contains functions with arbitrarily large distribution, provided that the ratio of this distribution to the distribution of a standard normal variable tends to zero. A similar result is also obtained for some classes of $\Lambda(2)$-spaces. Bibliography: 18 titles.

Основная цель обзора состоит в представлении результатов последнего десятилетия по описанию подпространств как $L_p$-пространств и пространств Орлича, так и общих симметричных пространств, порожденных независимыми функциями. Предлагается новый подход, основанный на использовании комбинации результатов теории симметричных пространств, методов теории...

The equivalence of the Haar system in a rearrangement invariant space \( X \) on \( [0,1] \) and a sequence of pairwise disjoint functions in some Lorentz space is known to imply that \( X=L_{2}[0,1] \) up to the equivalence of norms. We show that the same holds for the class of uniform disjointly homogeneous rearrangement invariant spaces and obta...

We study density estimates of an index set $\mathcal{A}$ under which the unconditionality (or even the weaker property of random unconditional divergence) of the corresponding Rademacher fractional chaos $\{r_{j_1}(t) \cdot r_{j_2}(t) \cdots r_{j_d}(t)\}_{(j_1,j_2,…,j_d) \in \mathcal{A}}$ in a symmetric space $X$ implies its equivalence in $X$ to t...

Замкнутая линейная оболочка функций Радемахера в пространстве $L^2[0,1]$ содержит функции со сколь угодно большим распределением при условии, что его отношение к распределению стандартной нормальной величины стремится к нулю. Аналогичный результат получен также для некоторых классов $\Lambda(2)$-пространств. Библиография: 18 названий.

Исследуются плотностные оценки индексного множества $\mathcal{A}$, при которых из безусловности (и даже случайной безусловной расходимости) дробного хаоса Радемахера $\{r_{j_1}(t)\cdot r_{j_2}(t)\cdots r_{j_d}(t)\}_{(j_1,j_2,…,j_d)\in \mathcal{A}}$ в симметричном пространстве $X$ вытекает его эквивалентность в $X$ каноническому базису в $\ell_2$. В...

From August 28 to August 30, 2023, the All-Russian scientific conference “Mathematics and Mathematical Modeling” was held at Samara University, organized by the Samara division of the regional scientific and educational mathematical center of the Volga Federal District

We investigate connections between upper/lower estimates for Banach lattices and the notion of relative s-decomposability, which has roots in interpolation theory. To get a characterization of relatively s-decomposable Banach lattices in terms of the above estimates, we assign to each Banach lattice X two sequence spaces XU and XL that are largely...

Subspaces of an Orlicz space LM generated by probabilistically independent copies of a function \(f \in {{L}_{M}}\), \(\int_0^1 {f(t){\kern 1pt} dt} = 0\), are studied. In terms of dilations of f, we get a characterization of strongly embedded subspaces of this type and obtain conditions that guarantee that the unit ball of such a subspace has equi...

We study density estimates of an index set $\mathcal{A}$, under which unconditionality (or even a weaker property of the random unconditional divergence) of the corresponding Rademacher fractional chaos $\{r_{j_1}(t)\cdot r_{j_2}(t)\cdot\dots\cdot r_{j_d}(t)\}_{(j_1,j_2,\dots,j_d)\in \mathcal{A}}$ in a symmetric space $X$ implies its equivalence in...

For a separable rearrangement invariant space X on [0, 1] of fundamental type we identify the set of all \(p\in [1,\infty ]\) such that \(\ell ^p\) is finitely represented in X in such a way that the unit basis vectors of \(\ell ^p\) (\(c_0\) if \(p=\infty \)) correspond to pairwise disjoint and equimeasurable functions. This can be treated as a fo...

We reformulate, modify and extend a comparison criteria of Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document} norms obtained by Nazarov–Podkorytov...

A subspace H of a rearrangement invariant space X on [0, 1] is strongly embedded in X if, in H, convergence in the X-norm is equivalent to convergence in measure. We obtain necessary and sufficient conditions on an Orlicz function M, under which the unit ball of an arbitrary strongly embedded subspace in the Orlicz space \(L_M\) has equi-absolutely...

It is shown that the space l_r is crudely finitely representable in the Lorentz space L_{p,q}[0, 1], 1 < p ≤ q < ∞, if and only if r = p or r = q. To the best of the author’s knowledge, this is the first example of a “natural” rearrangement-invariant space E on [0, 1] such that the set of all numbers r for which lr is crudely finitely representable...

We describe the spectral properties of the dilation operator \( Tx(t)=x(t/2) \), with \( t>0 \), in separable rearrangement invariant spaces of fundamental type. We show that these properties are determined by the values of the dilation indices of the spaces and apply the results to Orlicz spaces.

На основе изучения геометрических свойств безусловных квазибазисных последовательностей установлено, что в произвольном симметричном пространстве не существует безусловного квазибазиса, состоящего из неотрицательных функций. Кроме того, показано, что произвольная функциональная банахова решетка $X$, имеющая тип $p>1$, допускает введение такой эквив...

We establish Arazy–Cwikel type properties for the family of couples \( (\ell ^{p},\ell ^{q})\), \(0\le p<q\le \infty \), and show that \((\ell ^{p},\ell ^{q}) \) is a Calderón–Mityagin couple if and only if \(q\ge 1\). Moreover, we identify interpolation orbits of elements with respect to this couple for all p and q such that \(0\le p<q\le \infty \...

Показано, что пространство $l_r$ финитно грубо представимо в пространстве Лоренца $L_{p,q}[0,1]$, $1<p\leqslant q<\infty$, если и только если $r=p$ или $r=q$. Насколько нам известно, это первый пример "естественного" симметричного пространства $E$ на $[0,1]$, для которого множество всех $r$ таких, что $l_r$ финитно грубо представимо в $E$, не являе...

The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all K-monotone quasi-Banach lattices with respect to a L-convex quasi-Banach lattice couple have in fact a stronger property of the so-called K(p, q)-monotonicit...

Let |$({{\mathcal {F}}}_n)_{n\ge 0}$| be the standard dyadic filtration on |$[0,1)$|⁠. Let |${\mathbb {E}}_{{{\mathcal {F}}}_n}$| be the conditional expectation from |$ L_1=L_1[0,1)$| onto |${{\mathcal {F}}}_n$|⁠, |$n\ge 0$|⁠, and let |${\mathbb {E}}_{{{\mathcal {F}}}_{-1}} =0$|⁠. We present the sharp estimate for the distribution function of the m...

A subspace $H$ of a rearrangement invariant space $X$ on $[0,1]$ is strongly embedded in $X$ if, in $H$, convergence in $X$-norm is equivalent to convergence in measure. We obtain necessary and sufficient conditions on an Orlicz function $M$, under which the unit ball of an arbitrary strongly embedded subspace in the Orlicz space $L_M$ has equi-abs...

The main result of this paper establishes that the known Arazy-Cwikel property holds for classes of uniformly K-monotone spaces in the quasi-Banach setting provided that the initial couple is mutually closed. As a consequence, we get that the class of all quasi-Banach K-spaces (i.e., interpolation spaces which are described by the real K-method) wi...

For a separable rearrangement invariant space X on \((0,\infty )\) of fundamental type we identify the set of all \(p\in [1,\infty ]\) such that \(\ell ^p\) is finitely represented in X in such a way that the unit basis vectors of \(\ell ^p\) (\(c_0\) if \(p=\infty \)) correspond to pairwise disjoint and equimeasurable functions. This characterizat...

For a separable rearrangement invariant space $X$ on $[0,1]$ of fundamental type we identify the set of all $p\in [1,\infty]$ such that $\ell^p$ is finitely represented in $X$ in such a way that the unit basis vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint and equimeasurable functions. This can be treated as a follow up o...

For a separable symmetric sequence space X of fundamental type we identify the set F(X) of all p∈[1,∞] such that ℓp is block finitely represented in the unit vector basis {ek}k=1∞ of X in such a way that the unit basis vectors of ℓp (c0 if p=∞) correspond to pairwise disjoint blocks of {ek} with the same ordered distribution. It turns out that F(X)...

Let $(\mathcal{F}_n)_{n\ge 0}$ be the standard dyadic filtration on $[0,1]$. Let $\mathbb{E}_{\mathcal{F}_n}$ be the conditional expectation from $ L_1=L_1[0,1]$ onto $\mathcal{F} _n$, $n\ge 0$, and let $\mathbb{E}_{\mathcal{F} _{-1}} =0$. We present the sharp estimate for the distribution function of the martingale transform $T$ defined by \begin{...

The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all $K$-monotone quasi-Banach lattices with respect to a $L$-convex quasi-Banach lattice couple have in fact a stronger property of the so-called $K(p,q)$-monoto...

The main result of this paper establishes that the known Arazy-Cwikel property holds for classes of uniformly K-monotone spaces in the quasi-Banach setting provided that the initial couple is mutually closed. As a consequence, we get that the class of all quasi-Banach K-spaces (i.e., interpolation spaces which are described by the real K-method) wi...

Let be a nonseparable rearrangement-invariant space and let be the closure of the space of bounded functions in . Elements of orthogonal to , that is, elements , , such that for each , are investigated. The set of orthogonal elements is characterized in the case when is a Marcinkiewicz or an Orlicz space. If an Orlicz space is considered with the L...

Let f=∑k=0∞ckh2k, where {hn} is the classical Haar system, ck∈ℂ. Given a p∈(1,∞), we find the sharp conditions, under which the sequence {fn}n=1∞ of dilations and translations of f is a basis in the space Lp[0,1], equivalent to {hn}n=1∞. The results obtained depend substantially on whether p≥2 or 1<p<2 and include as the endpoints of the Lp-scale t...

Пусть $E$ - несепарабельное перестановочно-инвариантное пространство и $E_0$ - замыкание множества ограниченных функций в $E$. Работа посвящена изучению элементов пространства $E$, ортогональных подпространству $E_0$, т.е. таких $x\in E$, $x\ne 0$, что $\|x\|_{E} \le\|x+y\|_{E}$ для любого $y\in E_0$. Получена характеризация множества ортогональных...

It is well known that the span in \(L_p\) of a sequence of independent copies of a mean zero random variable \(f\in L_p\) is a subspace isomorphic to some Orlicz sequence space \(\ell _M\). It is also known (Astashkin et al. in Stud Math 230(1):41–57, 2015) that the distribution of such a random variable \(f\in L_p\) is essentially unique. We show...

It is well known that a Banach space need not contain any subspace isomorphic to a space ℓp (1 6 p ) or c0 (it was shown by Tsirelson in 1974). At the same time, by the famous Krivines theorem, every Banach space X always contains at least one of these spaces locally, i.e., there exist finite-dimensional subspaces of X of arbitrarily large dimensio...

We reformulate, modify and extend a comparison criteria of L p norms obtained by Nazarov-Podkorytov and place it in the general setting of interpolation theory and majorization theory. In particular, we give norm comparison criteria for general scales of interpolation spaces, including non-commutative L p and Lorentz spaces. As an application, we e...

Pure and Applied Functional Analysis, Volume 6, Number 3, 651-707, 2021

We establish Arazy-Cwikel type properties for the family of couples $(\ell^{p},\ell^{q})$, $0\le p<q\le\infty$, and show that $(\ell^{p},\ell^{q}) $ is a Calder\'on-Mityagin couple if and only if $q\ge1$. Moreover, we identify interpolation orbits of elements with respect to this couple for all $p$ and $q$ such that $0\le p<q\le\infty$ and obtain a...

Special issue of Pure and Applied Functional Analysis devoted to Extrapolation Theory http://www.ybook.co.jp/pafa.html

For a separable symmetric sequence space $X$ of fundamental type we identify the set ${\mathcal F}(X)$ of all $p\in [1,\infty]$ such that $\ell^p$ is block finitely represented in the unit vector basis $\{e_k\}_{k=1}^\infty$ of $X$ in such a way that the unit basis vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint blocks of...

For a separable rearrangement invariant space $X$ on $(0,\infty)$ of fundamental type we identify the set of all $p\in [1,\infty]$ such that $\ell^p$ is finitely represented in $X$ in such a way that the unit basis vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint and equimeasurable functions. This characterization hinges up...

Let \(1 \le p \le \infty \). A Banach lattice X is said to be p-disjointly homogeneous or \((p-DH)\) (resp. restricted \((p-DH)\)) if every normalized disjoint sequence in X (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in X to the unit vector basis of \(\ell _p\). We revisit DH-...

We obtain some new estimates that show the extremality of the Rademacher system in the set of sequences of independent functions considered in rearrangement invariant spaces.

We refine some earlier results by Flores, Hernández, Semenov, and Tradacete on compactness of the square of strictly singular endomorphisms and identifying general Banach lattices with the Kato property in the setting of rearrangement invariant spaces on [0, 1]. A Banach space X is said to have the Kato property if every strictly singular operator...

Получена характеризация дизъюнктно однородных функциональных пространств Орлича-Лоренца $\Lambda_{\varphi,w}$. С помощью нее найдены необходимые и достаточные условия, при которых в пространстве $\Lambda_{\varphi,w}$ выполняется аналог классической теорема Данфорда-Петтиса о равностепенной интегрируемости слабо компактных множеств в $L_1$. Показано...

Let l0 be the group (with respect to the coordinate-wise addition) of all sequences of real numbers x=(xk)k=1∞ that are eventually zero, equipped with the quasi-norm ‖x‖0=card{suppx}. A description of orbits of elements in the pair (l0,l1) is given, which complements (in the sequence space setting) the classical Calderón–Mityagin theorem on a descr...

Conditions under which the system of dilations and translations of a function f in a symmetric space X is a representing system in X are found. Previously a similar result was known only for the spaces Lp, 1 ⩽ p < ∞. In particular, each function f with \(\int_0^1 {f(t)dt \ne 0} \) in a Lorentz space Λϕ generates an absolutely representing system of...

So far, we studied the behaviour of the Rademacher functions in symmetric function spaces. In contrast to that, in the next three chapters we shall focus on Banach function spaces that are not symmetric. We start with consideration of spaces that do not have even the lattice property.

In this chapter, we study multiplicator spaces, generated by the Rademacher system in s.s.’s. They are called so because the elements of such a space \(\mathcal {M}(X)\) can be regarded as multiplicators acting from the Rademacher subspace of a given s.s. X into the whole space X.

It is well known that the norms of many Banach spaces, playing a significant role in functional analysis and its applications, are generated by positive sublinear operators and the Lp-norms. In particular, rather simple and both important spaces of such a type are the Cesàro spaces Cesp, 1 ≤ p ≤∞. In this chapter, we investigate the behaviour of th...

In this chapter we deal with Rademacher sums, whose coefficients are elements of a Banach space. In contrast to the scalar case, we have no longer the following “natural” orthogonality equation.

In Chap. 2, we have proved that in the case when a s.s. X contains the separable part G of the Orlicz space \(L_{N_2},\)\(N_2(t)=e^{t^2}-1,\) the Rademacher system is equivalent in X to the unit vector basis in ℓ2.

In the next two chapters we shall examine the Rademacher functions in general symmetric spaces (s.s.’s) on [0, 1] (see Appendix C). As we shall see, their behaviour in such a space X depends largely on how “near” X is located to the space L∞.

The Rademacher system takes a special place among all sequences of r.v.’s because of its very wide scope of applications. This is caused, in particular, by the fact that it might be an “extreme point” in one sense or another of a certain class of sequences. Thanks to that, the validity of some properties for systems from this class often follows im...

The main purpose of this chapter is comparing systems of measurable functions (or r.v.’s) with the “model” Rademacher system. To clarify the meaning of the word “comparison” , we introduce the following definitions.

Let a s.s. X contain the separable part G of the Orlicz space \(L_{N_2},\)\(N_2(u)=e^{u^2}-1\). According to Khintchine’s inequality (see Theorem 2.2), there exists a constant C = C(X) > 0 such that for every sequence \(a=(a_i)_{i=1}^\infty \in \ell _2\).

In what follows, \(\Delta _n^k\) are the dyadic subintervals of the interval [0, 1].

It is clear that the first-order Rademacher chaos is just the Rademacher system, and in the case of arbitrary order, this is a subsequence of the classical sequence of Walsh functions. In particular, system (6.1) is orthonormal on [0, 1] (see e.g. Corollary 1.2).

Let \(\{\mathcal {A}_k\}_{k=0}^\infty \) be a filtration of σ-algebras on a probability space \((\Omega ,\mathcal {A},\mathbb {P})\), i.e., \(\mathcal {A}_0\subset \mathcal { A}_1\subset \dots \subset \mathcal {A}_k\subset \dots \subset \mathcal {A}.\) A sequence of random variables (r.v.’s) \(\{v_k\}_{k=1}^\infty \) is said to be predictable with...

According to Khintchine’s inequality (see Theorem 1.4), for every p > 0 and all \(n\in \mathbb {N}\) there exist constants c > 0 and C > 0, which depend on p.

Another class of function spaces, whose norms are determined by the Lp-norms, is constituted by the Morrey spaces that have become the object of intensive research in recent decades.

When studying Rademacher sums with vector coefficients, often it is very useful to keep in mind that the norm of such a sum is nothing but as the supremum of a certain Bernoulli process.

By using results of the previous chapter, we obtain here necessary and sufficient conditions, under which an arbitrary sequence of r.v.’s contains a subsequence dominated in distribution by (equivalent in distribution to) the Rademacher system on the interval [0, 1].

We characterize the Garsia-Rodemich spaces associated with a rearrangement invariant space via local maximal operators. Let $Q_{0}$ be a cube in $R^{n}$. We show that there exists $s_{0}\in(0,1),$ such that for all $0<s<s_{0},$ and for all r.i. spaces $X(Q_{0}),$ we have% \[ GaRo_{X}(Q_{0})=\{f\in L^{1}(Q_{0}):\Vert f\Vert_{GaRo_{X}}\simeq\Vert M_{...

Let $E(0,1)$ be a symmetric space on $(0,1)$ and $C_F$ be a symmetric ideal of compact operators on the Hilbert space $\ell_2$ associated with a symmetric sequence space $F$. We give several criteria for $E(0,1)$ and $ F$ so that $E(0,1)$ does not embed into the ideal $C_F$, extending the result for the case when $E(0,1)=L_p(0,1)$ and $F=\ell_p $,...

Let $E$ be a rearrangement invariant (r.i.) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\infty)$ such that $f^*\chi_{[0,1]}\in E$ and $f^*\chi_{[1,\infty)}\in L^2$. We reveal close connections between properties of the generalized Rosenthal's space, corresponding to the space $Z_E$, and the behaviour of i...

Let $f=\sum_{k=0}^{\infty}c_kh_{2^k}$, where $\{h_n\}$ is the classical Haar system, $c_k\in\mathbb{C}$. Given a $p\in (1,\infty)$, we find the sharp conditions, under which the sequence $\{f_n\}_{n=1}^\infty$ of dilations and translations of $f$ is a basis in the space $L^p[0,1]$, equivalent to $\{h_n\}_{n=1}^\infty$. The results obtained depend s...

Let $1\le p\le\infty$. A Banach lattice $X$ is said to be $p$-disjointly homogeneous or $(p-DH)$ (resp. restricted $(p-DH)$) if every normalized disjoint sequence in $X$ (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in $X$ to the unit vector basis of $\ell_p$. We revisit $DH$-pro...

Let $l_0$ be the group (with respect to the coordinate-wise addition) of all sequences of real numbers $x=(x_k)_{k=1}^\infty$ that are eventually zero, equipped with the quasi-norm $\|x\|_0={\rm card}\{supp\,x\}$. A description of orbits of elements in the pair $(l_0,l_1)$ is given, which complements (in the sequence space setting) the classical Ca...

We discuss some aspects of Extrapolation Theory. The presentation includes many examples and open problems.

We study the family of rearrangement invariant spaces E containing subspaces on which the E-norm is equivalent to the L1-norm and a certain geometric characteristic related to the Kadec–Pełcziński alternative is extremal. We prove that, after passing to an equivalent norm, any space with nonseparable second Köthe dual belongs to this family. In the...

Изучается семейство симметричных пространств $E$, содержащих подпространства, на которых эквивалентны нормы $E$ и $L_1$, а также экстремальна одна геометрическая характеристика, связанная с альтернативой Кадеца-Пелчинского. Доказано, что этому семейству после эквивалентной перенормировки принадлежит любое пространство, имеющее несепарабельное второ...

Answering to a recent question raised by Le\'{s}nik, Maligranda, and Tomaszewski, we prove that there is an Orlicz function $\Phi$ with the upper Matuszewska-Orlicz index equal to $1$ such that the Orlicz space $L_\Phi$ does not satisfy Dunford-Pettis criterion of weak compactness.

Найдены условия, при которых последовательность сжатий и сдвигов функции $f$ из симметричного пространства $X$ является представляющей системой в $X$. Ранее подобный результат был известен только в случае пространства $L_p$, $1\le p<\infty$. В частности, для того, чтобы каждая функция $f$ из пространства Лоренца $\varLambda_{\varphi}$, $\int_0^1f(t...

Let X be an arbitrary separable symmetric function space on [0, 1]. By using a combination of the frame approach and the notion of the multiplicator space \(\mathscr {M}(X)\) of X with respect to the tensor product, we investigate the problem when the sequence of dyadic dilations and translations of a function \(f\in X\) is a representing system in...

This book presents a systematic treatment of the Rademacher system, one of the most important unifying concepts in mathematics, and includes a number of recent important and beautiful results related to the Rademacher functions. The book discusses the relationship between the properties of the Rademacher system and geometry of some function spaces....

Let and be rearrangement invariant spaces on , and let . This embedding is said to be strict if the functions in the unit ball of the space have absolutely equicontinuous norms in . For the main classes of rearrangement invariant spaces necessary and sufficient conditions are obtained for an embedding to be strict, and also the relationships this c...

Пусть $E$ и $F$ - перестановочно-инвариантные пространства на $[0,1]$, $E\subset F$. Это вложение называется строгим, если функции из единичного шара пространства $E$ имеют равностепенно абсолютно непрерывные нормы в $F$. Получены необходимые и достаточные условия строгости вложения для основных классов перестановочно-инвариантных пространств, а та...