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Grigori A. Karagulyan

Publications and source records attributed to Grigori A. Karagulyan.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative estimates for the absolute convergence of wavelet-type series

We establish new quantitative estimates for general systems of functions with wavelet-type dyadic structure. These estimates are applied to obtain the optimal growth of various types of Weyl multipliers for certain wavelet-type systems. Some of our results are sufficiently general to allow the orthogonality assumption to be removed. In particular, as a consequence of these estimates we show that the condition \begin{equation*} \sum_{n=1}^\infty\frac{1}{nw(n)}<\infty \end{equation*} is necessary and sufficient for an increasing sequence $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for an arbitrary wavelet-type system. We also prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any rearranged wavelet-type system, and that this bound is optimal.

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arXiv2026

On UC-multipliers for multiple trigonometric systems

We investigate the class of sequences $w(n)$ that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds $\log n\lesssim w(n)\lesssim\log^2 n$. Our main result establishes a general equivalence principle between one-dimensional and multidimensional trigonometric systems, which allows one to extend certain estimates known for the one-dimensional case to higher dimensions.

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arXiv2025

Bounded oscillation operators on BMO spaces

Bounded Oscillation (BO) operators were recently introduced in the author's paper [13], where it was proved that many operators in harmonic analysis (Calderón-Zygmund operators, Carleson type operators, martingale transforms, Littlewood-Paley square functions, maximal operators, etc) are $BO$ operators. $BO$ operators are defined on abstract measure spaces equipped with a basis of abstract balls. The abstract balls in their definition owe four basic properties of classical balls in $\mathbb{R}^n$, which are crucial in the study of singular operators on $\mathbb{R}^n$. Among various properties studied in these papers it was proved that $BO$ operators allow pointwise sparse domination, establishing the $A_2$-conjecture for those operators. In the present paper we study boundedness properties of $BO$ operators on $BMO$ spaces. In particular, we prove that general $BO$ operators boundedly map $L^\infty$ into $BMO$, and under a logarithmic localization condition those map $BMO$ into itself. We obtain these properties as corollaries of new local type bounds, involving oscillations of functions over the balls. We apply the results in the $BMO$ estimations of Calderón-Zygmund operators, martingale transforms, Carleson type operators, as well as in the unconditional basis properties of general wavelet type systems in atomic Hardy spaces $H^1$.

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arXiv2025

Maximal operators on spaces BMO and BLO

We consider maximal kernel-operators on abstract measure spaces $(X,μ)$ equipped with a ball-basis. We prove that under certain asymptotic condition on the kernels those operators maps boundedly BMO(X) into BLO(X), generalizing the well-known results of Bennett-DeVore-Sharpley and Bennett for the Hardy-Littlewood maximal function. As a particular case of such an operator one can consider the maximal function \begin{equation} M_ϕf(x)=\sup_{r>0}\frac{1}{r^d}\int_{R^d}|f(t)|ϕ\left(\frac{x-t}{r}\right)dt, \end{equation} and its non-tangential version. Here $ϕ(x)\ge 0$ is a bounded spherical function on $R^d$, decreasing with respect to $|x|$ and satisfying the bound \begin{equation*} \int_{R^d}ϕ(x)\log (2+|x|)dx<\infty. \end{equation*} We prove that if $f\in BMO(R^d)$ and $M_ϕ(f)$ is not identically infinite, then $M_ϕ(f)\in BLO(R^d)$. Our main result is an inequality, providing an estimation of certain local oscillation of the maximal function $M(f)$ by a local sharp function of $f$.

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arXiv2024

On the best constants in Khintchine type inequalities for martingales

For discrete martingale-difference sequences $d=\{d_1,\ldots,d_n\}$ we consider Khintchine type inequalities, involving certain square function $\mathfrak S (d)$ considered by Chang-Wilson-Wolff in 1982. In particular, we prove \begin{equation} \left\|\sum_{k=1}^nd_k\right\|_p\le 2^{1/2}\big(Γ((p+1)/2))/\sqrtπ\big)^{1/p}\|\mathfrak S(d)\|_\infty,\quad p\ge 3, \end{equation} where the constant on the right hand side is the best possible and the same as known for the Rademacher sums $\sum_{k=1}^na_kr_k$. Moreover, for a fixed $n$ the constant in the inequality can be replaced by $\sum_{k=1}^nr_k/\sqrt{n}$. We apply a technique, reducing the general case to the case of Haar and Rademacher sums, that allows also establish a sub-Gaussian estimate \begin{equation} {\bf E}\left[\exp\left(λ\cdot \left(\frac{\sum_{k=1}^nd_k}{\|\mathfrak S(d)\|_\infty}\right)^2\right)\right]\le \frac{1}{\sqrt{1-2λ}},\quad 0<λ<1/2, \end{equation} where the constant on the right hand side is the best possible.

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arXiv2023

On the convergence sets of operator sequences on spaces of homogeneous type

We consider sequences of operators $U_n:L^1(X)\to M(X)$, where $X$ is a space of homogeneous type. Under certain conditions on the operators $U_n$ we give a complete characterization of convergence (divergence) sets of functional sequences $U_n(f)$, where $f\in L^p(X)$, $1\le p\le \infty$. The results are applied to characterize convergence sets of some specific operator sequences in classical analysis.

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arXiv2023

New estimates for bounded oscillation operators

In this paper we consider a generalized version of bounded oscillation operators, involving new parameters in the definition, as well as considering the operators on vector-valued function spaces. With this definition we will capture some more operators to the class of Bounded oscillation operators. We prove sparse domination and exponential decay estimates, with various applications in harmonic analysis operators. We provide a new simplified approach, separating certain set theoretic proposition, which become a basic tool in the proofs of the main results. For a "narrowed" class of bounded oscillation operators we also obtain new type of sparse estimation, involving mean oscillation instead of integral averages in the definition of sparse operators. Among with new corollaries we recover also series of results obtained in recent years. In particular, we prove the boundedness of maximally modulated Calderón-Zygmund operators on spaces BMO.

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arXiv2022

Sharp inequalities involving multiplicative chaos sums

The present note is an essential addition to the author's arxiv paper arXiv:2001.01070, concerning general multiplicative systems of random variables. Using some lemmas and the methodology of \cite{Kar4}, we obtain a general extreme inequality, with corollaries involving Rademacher chaos sums and those analogues for multiplicative systems. In particular we prove that a system of functions generated by bounded products of a multiplicative system is a convergence system.

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arXiv2022

Asymptotic estimates for double-coverings

A collection of finite sets $\{A_1, A_2,\ldots, A_{p}\}$ is said to be a double-covering if each $a\in \cup_{k=1}^{p}A_k$ is included in exactly two sets of the collection. For fixed integers $l$ and $p$, let $μ_{l,p}$ be the number of equivalency classes of double-coverings with $\#(A_k)=l$, $k=1,2,\ldots,p$. We characterize the asymptotic behavior of the quantity $μ_{l,p}$ as $p\to \infty$. The results are applied to give an alternative approach to the Bonami-Kiener hypercontraction inequality.

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arXiv2022

On the convergence of multiple ergodic means

Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,μ)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots s_{n}}\sum_{j_1=0}^{s_1-1}\cdots\sum_{j_n=0}^{s_n-1}f\left(U_1^{j_1}\cdots U_n^{j_n} x \right) \end{equation} as $\min_j s_j\to\infty $, for any function $f\in L\log^{d-1}(X)$, where $d\le n$ is the rank of the transformations. The result gives a generalization of a theorem by N. Dunford and A. Zygmund, claiming the convergence of the means in a narrower class of functions $L\log^{n-1}(X)$.

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arXiv2022

On generalized lacunary series

Given lacunary sequence of integers, $n_k$, $n_{k+1}/n_k>λ>1$, we define a new sequence $\{m_k\}$ formed by all possible $l$-wise sums $\pm n_{k_1}\pm n_{k_2}\pm \ldots\pm n_{k_l}$. We prove if $λ>λ_l$, then any series \begin{equation} \sum_kc_ke^{im_kx},\qquad (1) \end{equation} with $\sum_k|c_k|^2<\infty$ converges almost everywhere after any rearrangement of the terms, where $1<λ_l<2$ is a certain critical value. We establish this property, proving a new Khintchine type inequality $\|S\|_p\le C_{l,λ,p}\|S\|_2$, $p>2$, where $S$ is a finite sum of form (1). For $λ\ge 3$, we also establish a sharp rate $p^{l/2}$ for the growth of the constant $C_{l,λ,p}$ as $p\to\infty$. Such an estimate for the Rademacher chaos sums was proved independently by Bonami and Kiener. In the case of $λ\ge 3$ we also establish some inverse convergence properties of series (1): 1) if series (1) converges a.e., then $\sum_k|c_k|^2<\infty$, 2) if it a.e. converges to zero, then $c_k=0$.

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arXiv2021

On wavelet polynomials and Weyl multipliers

For the wavelet type orthonormal systems $ϕ_n$, we establish a new bound \begin{equation} \left\|\max_{1\le m\le n}\left|\sum_{j\in G_m}\langle f,ϕ_j\rangle ϕ_j\right|\right\|_p\lesssim \sqrt{\log (n+1)}\cdot \|f\|_p,\quad 1<p<\infty, \end{equation} where $G_m\subset N$ are arbitrary sets of indexes. Using this estimate, we prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any orthonormal system of non-overlapping wavelet polynomials. It will also be remarked that $\log n$ is the optimal sequence in this context.

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arXiv2020

On Weyl multipliers of the rearranged trigonometric system

We prove that the condition \begin{equation} \sum_{n=1}^\infty\frac{1}{nw(n)}<\infty \end{equation} is necessary for an increasing sequence of numbers $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for the trigonometric system. This property for Haar, Walsh, Franklin and some other classical orthogonal systems was known long ago. The proof of this result is based on a new sharp logarithmic lower bound on $L^2$ of the majorant operator related to the rearranged trigonometric system.

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arXiv2019

On good-$λ$ inequalities for couples of measurable functions

We give a domination condition implying good-$λ$ and exponential inequalities for couples of measurable functions. Those inequalities recover several classical and new estimations involving some operators in Harminic Analysis. Among other corollaries we prove a new exponential estimate for Carleson operators. The main results of the paper are considered in a general setting, namely, on abstract measure spaces equipped with a ball-basis.

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arXiv2019

Menshov type correction theorems for sequences of compact operators

We prove Menshov type "correction" theorems for sequences of compact operators, recovering several results of Fourier series in trigonometric and Walsh systems. The paper clarifies the main ingredient, which is important in the study of such "correction" theorems. That is the weak-$L^1$ estimate for the maximal Fourier sums of indicator functions of some specific sets.

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