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Sergey Astashkin

Publications and source records attributed to Sergey Astashkin.

9 recordsLinked to original sources
arXiv2022

Optimal range of Haar martingale transforms and its applications

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{align*} Tf=\sum_{m=0}^\infty \left( \mathbb{E}_{\mathcal{F}_{2m}} f-\mathbb{E}_{\mathcal{F}_{2m-1}}f \right), ~f\in L_1, \end{align*} in terms of the classical Calderón operator. As an application, for a given symmetric function space $E$ on $[0,1]$, we identify the symmetric space $\mathcal{S}_E$, the optimal Banach symmetric range of martingale transforms/Haar basis projections acting on $E$.

math.PR↗
arXiv2021

A characterization of $\ell^p$-spaces symmetrically finitely represented in symmetric sequence spaces

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 $\{e_k\}$ with the same ordered distribution. It turns out that ${\mathcal F}(X)$ coincides with the set of approximate eigenvalues of the operator $(x_k)\mapsto \sum_{k=2}^\infty x_{[k/2]}e_k$ in $X$. In turn, we establish that the latter set is the interval $[2^{α_X},2^{β_X}]$, where $α_X$ and $β_X$ are the Boyd indices of $X$. As an application, we find the set ${\mathcal F}(X)$ for arbitrary Lorentz and separable sequence Orlicz spaces.

math.FA↗
arXiv2020

Disjointly homogeneous Orlicz spaces revisited

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 $DH$-properties of Orlicz spaces and refine some previous results of this topic, showing that $(p-DH)$-property is not stable in the class of Orlicz spaces and the classes of restricted $(p-DH)$ and $(p-DH)$ Orlicz spaces are different. Moreover, we give a characterization of uniform $(p-DH)$ Orlicz spaces and establish also closed connections between this property and the duality of $DH$-property.

math.FA↗
arXiv2020

A version of Calderón-Mityagin theorem for the class of rearrangement invariant groups

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 Calderón-Mityagin theorem on a description of orbits of elements in the pair $(l_1,l_\infty)$. As a consequence, we obtain that the pair $(l_0,l_1)$ is ${\mathcal K}$-monotone.

math.FA↗
arXiv2019

Garsia-Rodemich spaces: Local Maximal Functions and Interpolation

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% \[ GaRo_{X}(Q_{0})=\{f\in L^{1}(Q_{0}):\Vert f\Vert_{GaRo_{X}}\simeq\Vert M_{0,s,Q_{0}}^{\#}f\Vert_{X}<\infty\}, \] where $M_{0,s,Q_{0}}^{\#}$ is the Strömberg-Jawerth-Torchinsky local maximal operator. Combined with a formula for the $K-$functional of the pair $(L^{1},BMO)$ obtained by Jawerth-Torchinsky, our result shows that the $GaRo_{X}$ spaces are interpolation spaces between $L^{1}$ and $BMO.$ Among the applications, we prove, using real interpolation, the monotonicity under rearrangements of Garsia-Rodemich type functionals. We also give an approach to Sobolev-Morrey inequalities via Garsia-Rodemich norms, and prove necessary and sufficient conditions for $GaRo_{X}(Q_{0})=X(Q_{0}).$ Using packings, we obtain a new expression for the $K-$functional of the pair $(L^{1},BMO)$.

math.FA↗
arXiv2018

$L_p +L_q$ and $L_p \cap L_q$ are not isomorphic for all $1 \leq p, q \leq \infty$, $p \neq q$

We prove that if $1 \leq p, q \leq \infty$, then the spaces $L_p +L_q$ and $L_p \cap L_q$ are isomorphic if and only if $p = q$. In particular, $L_2 +L_{\infty}$ and $L_2 \cap L_{\infty}$ are not isomorphic which is an answer to a question formulated in the paper S. V. Astashkin and L. Maligranda, \textit{$L_p + L_{\infty}$ and $L_p \cap L_{\infty}$ are not isomorphic for all $1 \leq p < \infty, p \neq 2$}, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2181--2194.

math.FA↗
arXiv2015

Rademacher functions in weighted symmetric spaces

The closed span of Rademacher functions is investigated in the weighted spaces X(w), where X is a symmetric space on [0,1] and w is a positive measurable function on [0,1]. By using the notion and properties of the Rademacher multiplicator space of a symmetric space, we give a description of the weights w for which the Rademacher orthogonal projection is bounded in X(w).

math.FA↗
arXiv2014

Disjointly homogeneous rearrangement invariant spaces via interpolation

A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of l_p. Employing methods from interpolation theory, we clarify which rearrangement invariant (r.i.) spaces on [0,1] are p-disjointly homogeneous. In particular, for every 1<p< infty and any increasing concave function f on [0,1], which is not equivalent neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function f. Moreover, in the class of all interpolation r.i. spaces with respect to the Banach couple of Lorentz and Marcinkiewicz spaces with the same fundamental function, dilation indices of which are non-trivial, for every 1<p< infty, there is only a unique p-disjointly homogeneous space.

math.FA↗