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G. A. Karagulyan

Publications and source records attributed to G. A. Karagulyan.

8 recordsLinked to original sources
arXiv2016

On Riemann sums and maximal functions in $\ZR^n$

In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums \md0 R_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f\bigg(x+\frac{k}{n}\bigg),\quad x\in \ZT. \emd We establish a relevant connection between Riemann and ordinary maximal functions, which allows to use techniques and results of the theory of differentiations of integrals in $\ZR^n$ in mentioned problems. In particular, we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums $R_{n_k}f(x)$ converge almost everywhere for any $f\in L^p$ with $p>1$.

math.CA↗
arXiv2013

On generalizations of Fatou's theorem for the integrals with general kernels

We define $λ(r)$-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson-Stiltjes integrals for general kernels $\{φ_r\}$, forming an approximation of identity. We prove that the bound \md0 \limsup_{r\to 1}λ(r) \|φ_r\|_\infty<\infty \emd is necessary and sufficient for almost everywhere $λ(r)$-convergence of the integrals \md0 \int_\ZT φ_r(t-x)dμ(t). \emd

math.CA↗
arXiv2013

On a theorem of Littlewood

In 1927 Littlewood constructed an example of bounded holomorphic function on the unit disk, which diverges almost everywhere along rotated copies of any given curve in the unit disk ending tangentially to the boundary. This theorem was the complement of a positive theorem of Fatou 1906, establishing almost everywhere nontangential convergence of bounded holomorphic functions. There are several generalizations of the Littlewood's theorem which proofs are based on the specific properties of Poisson kernel. We generalize Littlewood's theorem for operators having general kernels.

math.CA↗
arXiv2012

On the sweeping out property for convolution operators of discrete measures

Let $μ_n$ be a sequence of discrete measures on the unit $\ZT=\ZR/\ZZ$ with $μ_n(0)=0$, and $μ_n((-δ,δ))\to 1$, as $n\to\infty$. We prove that the sequence of convolution operators $(f\astμ_n)(x)$ is strong sweeping out, i.e. there exists a set $E\subset\ZT$ such that \md0 \lim\sup_{n\to\infty}(\ZI_E\astμ_n)(x)= 1,\quad \lim\inf_{n\to\infty}(\ZI_E\astμ_n)(x)= 0, \emd almost everywhere on $\ZT$.

math.CA↗
arXiv2009

Divergence of general localized operators on the sets of measure zero

We consider sequences of linear operators $U_nf(x)$ with localization property. It is proved that for any set $E$ of measure zero there exists a set $G$ for which $U_n\ZI_G(x)$ diverges at each point $x\in E$. This result is a generalization of analogous theorems known for the Fourier sums operators with respect to different orthogonal systems.

math.CA↗
arXiv2009

On the complete characterization of differentiation sets of integrals

Let $B_θ$ be the family of rectangles in the plane $R^2$, having slope $θ$ with the abscissa. We say a set of slopes $Θ$ is $D$-set if there exists a function $f\in L(R^2)$, such that the basis $B_θ$ differentiates integral of $f$ if $θ\not\inΘ$ and $\bar D_θf(x)=\infty $ almost everywhere if $θ\inΘ$. If the condition $\bar D_θf(x)=\infty $ holds on a set of positive measure (instead of a.e.) we shall say it is $WD$-set. It is proved, that $Θ$ is $D$-set($WD$-set) if and only if it is $G_δ$($G_{δσ}$).

math.CA↗
arXiv2005

On directional maximal operators associated with generalized lacunary sets

Let $Ω$ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_Ωf(x)=\sup_{δ>0, ω\in Ω} \frac{1}{2δ}\int_{-δ}^δ|f(x+tω)|dt. \emd for the generalized lacunary sets $Ω$ associated with an integer $μ>0$. It is proved the following sharp inequality: $$ \|M_Ωf(x)\|_2\lesssim \sqrtμ \|f\|_2. $$

math.CA↗