arXiv subjects

Sergey V. Astashkin

Publications and source records attributed to Sergey V. Astashkin.

At least 19 recordsLinked to original sources

arXiv2026

Rigidity of sets of independent functions in symmetric spaces

We say that a symmetric function space $X$ has the $(IR)$ property whenever all sets of $N$ independent mean zero functions $f_1,\ldots,f_N\in X$, $\|f_k\|_X\ge 1$, are poorly approximated by any linear combinations of arbitrary $n$ functions, if $n$ is sufficienly smaller that $N$; namely, for some $γ=γ(X)>0$ we have $d_n(\{f_1,\ldots,f_N\},X)\ge γ$, $n\le γN$, where $d_n(K,X)$ is the Kolmogorov $n$-width of the set $K\subset X$. The spaces $X=L_p$ satisfy this property if and only if $1\le p\le2$ or $p=\infty$. The goal of this paper is to move from $L_p$ scale to a larger class of symmetric spaces. We obtain rather broad conditions, under which such a space $X$ has the $(IR)$ property and prove precise statements for particular scales of Lorentz $L_{p,q}$ spaces and Orlicz spaces.

math.FA↗

arXiv2026

Optimal upper and lower sequence spaces with applications

We study the optimal upper $X_U$ and lower $X_L$ sequence spaces that can be assigned to each Banach lattice $X$. These spaces are symmetric, have the Fatou property and the unit vector basis has in these spaces very special properties. Determined by the order structure of $X$ the spaces $X_U$ and $X_L$ turn out to be very useful when studying Banach lattices. Among other results, in terms of these constructions, we identify Banach lattices that satisfy equal-norm upper and lower $p$-estimates, give a characterization of $L_p(μ)$-spaces, derive some properties of the tensor product operator in Lorentz and Orlicz spaces, identify Orlicz spaces in which the unit vector basis is upper (resp. lower) semi-homogeneous.

math.FA↗

arXiv2025

On extreme points of the unit ball of a Hardy-Lorentz space

We investigate the problem of a characterization of extreme points of the unit ball of a Hardy-Lorentz space $H(Λ(φ))$, posed by Semenov in 1978. New necessary and sufficient conditions, under which a normalized function $f$ in $H(Λ(φ))$ belongs to this set, are found. The most complete results are obtained in the case when $f$ is the product of an outer analytic function and a Blaschke factor.

math.FA↗

arXiv2025

Looking for a continuous version of Bennett--Carl theorem

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 summing for each $p<q<2$, where $X_p$ is the unique r.i. Banach function space in which the Rademacher system spans copy of $l^p$. This result may be regarded as a continuous version of the well-known Carl--Bennett theorem. Two different approaches to the problem and extensive discussion on them are presented. We also conclude summability type of a kind of Sobolev embedding in the critical case.

math.FA↗

arXiv2024

Random unconditional convergence of Rademacher chaos in $L_\infty$ and sharp estimates for discrepancy of weighted graphs and hypergraphs

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 establish sharp two-sided estimates for the discrepancy of edge-weighted graphs and hypergraphs. Some of these results extend the classical theorem proved by Erdös and Spencer for the unweighted case.

math.PR↗

arXiv2024

On subspaces of Orlicz spaces spanned by independent copies of a mean zero function

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 $\ell_ψ$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions, guaranteeing that the unit ball of such a subspace consists of functions with equicontinuous norms in $L_M$, are found. In particular, we prove that there is a wide class of Orlicz spaces $L_M$ (containing $L^p$-spaces, $1\le p< 2$), for which each of the above properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $ψ$ satisfy the inequality: $α_ψ^0>β_M^\infty$.

math.FA↗

arXiv2024

On the set of extreme points of the unit ball of a Hardy-Lorentz space

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 $Λ(φ)$ on $[0,a]$ whenever $φ$ is a not linear, strictly increasing, concave, continuous function on $[0,a]$ with $φ(0)=0$. As a consequence, we complement the classical de Leeuw-Rudin theorem on a description of extreme points of the unit ball of $H^1$ showing that $H^1$ is a unique Hardy-Lorentz space $H(Λ(φ))$, for which every extreme point of the unit ball is a normed outer function. Moreover, assuming that $φ$ is strictly increasing and strictly concave, we prove that every function $f\in H(Λ(φ))$, $\|f\|_{H(Λ(φ))}=1$, such that the absolute value of its nontangential limit ${f}(e^{it})$ is a constant on some set of positive measure of $[0,2π]$, is an extreme point of the unit ball of $H(Λ(φ))$.

math.FA↗

arXiv2023

S-Decomposable Banach Lattices, Optimal Sequence Spaces and Interpolation

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 determined by the set of p, for which lp is finitely lattice representable in X. As an application, we obtain an orbital factorization of relative K-functional estimates for Banach couples (X0, X1) and (Y0, Y1) through some suitable couples of weighted Lp-spaces provided if Xi, Yi are relatively s-decomposable for i = 0, 1. Also, we undertake a detailed study of the properties of optimal upper and lower sequence spaces XU and XL, and, in particular, prove that these spaces are rearrangement invariant. In the Appendix, a description of the optimal upper sequence space for a separable Orlicz space as a certain intersection of some special Musielak-Orlicz sequence spaces is given

math.FA↗

arXiv2022

Symmetric finite representability of $\ell^p$-spaces in rearrangement invariant spaces on $[0,1]$

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 of a paper by the first-named author related to separable rearrangement invariant spaces on $(0,\infty)$.

math.FA↗

arXiv2021

Arazy-Cwikel and Calderón-Mityagin type properties of the couples $(\ell^{p},\ell^{q})$, $0 \le p<q\le\infty$

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 $q\ge1$. 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 simple positive solution of a Levitina-Sukochev-Zanin problem, clarifying its connections with whether $(\ell^{p},\ell^{q})$ has the Calderón-Mityagin property or not.

math.FA↗

arXiv2021

A description of interpolation spaces for quasi-Banach couples by real $K$-method

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 $K(p,q)$-monotonicity for some $0<q\leq p\leq 1$, which allows us to get their description by the real $K$-method. Moreover, we obtain a refined version of the $K$-divisibility property for Banach lattice couples and then prove an appropriate version of this property for $L$-convex quasi-Banach lattice couples. The results obtained are applied to refine interpolation properties of couples of sequence $l^{p}$- and function $L^{p}$-spaces, considered for the full range $0<p<\infty $.

math.FA↗

arXiv2021

Arazy-Cwikel property for quasi-Banach couples

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) with respect to an arbitrary mutually closed Banach couple enjoys the Arazy-Cwikel property. Another consequence complements some previous results by Bykov and Ovchinnikov, showing that this property holds also for the class of all interpolation quasi-Banach spaces with respect to a quasi-Banach couple whenever all the couples involved have the uniform Calderón-Mityagin property. We apply these results to some classical families of spaces.

math.FA↗

arXiv2021

Majorization revisited: Comparison of norms in interpolation scales

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 extend the classical Ball's integral inequality, which lies at the basis of his famous result on sections of the $n-$dimensional unit cube.

math.FA↗

arXiv2020

Rosenthal's space revisited

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 $f^*χ_{[0,1]}\in E$ and $f^*χ_{[1,\infty)}\in L^2$. We reveal close connections between properties of the generalized Rosenthal's space, corresponding to the space $Z_E$, and the behaviour of independent symmetrically distributed random variables in $E$. The results obtained are applied to consider the problem of the existence of isomorphisms between r.i.\ spaces on $[0,1]$ and $(0,\infty)$. Exploiting particular properties of disjoint sequences, we identify a rather wide new class of r.i.\ spaces on $[0,1]$ ``close'' to $L^\infty$, which fail to be isomorphic to r.i.\ spaces on $(0,\infty)$. In particular, this property is shared by the Lorentz spaces $Λ_2(\log^{-α}(e/u))$, with $0<α\le 1$.

math.FA↗

arXiv2020

Sequences of dilations and translations equivalent to the Haar system in $L^p$-spaces

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 substantially on whether $p\ge 2$ or $1<p<2$ and include as the endpoints of the $L_p$-scale the spaces $BMO_d$ and $H_d^1$. The proofs are based on an appropriate splitting the set of positive integers $\mathbb{N}=\cup_{d=1}^\infty N_d$ so that the equivalence of $\{f_n\}_{n=1}^\infty$ to the Haar system in $L_p$ would be ensured by the fact that $\{f_n\}_{n\in N_d}$ is a basis in the subspace $[h_{m},m\in N_d]_{L_p}$, equivalent to the Haar subsequence $\{h_n\}_{n\in N_d}$ for every $d=1,2,\dots$.

math.FA↗

arXiv2020

On one class of Orlicz functions

Answering to a recent question raised by Leśnik, Maligranda, and Tomaszewski, we prove that there is an Orlicz function $Φ$ with the upper Matuszewska-Orlicz index equal to $1$ such that the Orlicz space $L_Φ$ does not satisfy Dunford-Pettis criterion of weak compactness.

math.FA↗

arXiv2019

Representing systems of dilations and translations in symmetric spaces

Let $X$ be an arbitrary separable symmetric 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 the space $X$. The main result reads that this holds whenever $\int_0^1 f(t)\,dt\ne 0$ and $f\in \mathscr{M}(X)$. Moreover, the condition $f\in\mathscr{M}(X)$ turns out to be sharp in a certain sense. In particular, we prove that a decreasing nonnegative function $f$, $f\ne 0$, from a Lorentz space $\varLambda_φ$ generates an absolutely representing system of dyadic dilations and translations in $\varLambda_φ$ if and only if $f\in\mathscr{M}(\varLambda_φ)$.

math.FA↗

arXiv2019

Duality problem for disjointly homogeneous rearrangement invariant spaces

Let $1\le p<\infty$. A Banach lattice $E$ is said to be disjointly homogeneous (resp. $p$-disjointly homogeneous) if two arbitrary normalized disjoint sequences from $E$ contain equivalent in $E$ subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in $E$ to the unit vector basis of $l_p$). Answering a question raised in 2014 by Flores, Hernandez, Spinu, Tradacete, and Troitsky, for each $1<p<\infty$, we construct a reflexive $p$-disjointly homogeneous rearrangement invariant space on $[0,1]$ whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to $l_p$, $1\le p<\infty$, or $c_0$.

math.FA↗