G. A. Karagulyan

On unboundedness of maximal operators for directional Hilbert transforms

. We show that for any infinite set of unit vectors U in R 2 the maximal operator defined by H u f(x) = supu∈U|pv∞-∞ f(x-tu) t dt| x∈R 2 , is not bounded in L 2 (R 2 ).

On complete characterization of divergence sets of fourier-haar series

The paper proves that E ⊂ [0, 1] is a set of divergence points of Fourier-Haar series of a function f ∈ L∞[0, 1] if and only if E is a Gδσtype set of zero measure.

On equivalency of martingales and related problems

We study some almost everywhere convergence problems for martingales. We establish various equivalency theorems, which show that in some problems of martingales theory the general martingales can be replaced by Haar martingales. Some applications of the obtained results to the theory of differentiation of integrals and convergence of Riemann sums are also discussed.

Exponential Estimates of the Calderón--Zygmund Operator and Related Questions about Fourier Series

In this paper we study the properties of the maximal operator generated by the Calderón--Zygmund operator. In particular, we refine Hunts inequality.

On the divergence of triangular and eccentric spherical sums of double Fourier series

We construct a continuous function on the torus with almost everywhere divergent triangular sums of double Fourier series. We also prove an analogous theorem for eccentric spherical sums. Bibliography: 14 titles.

Divergent triangular sums of double trigonometric Fourier series

In this paper we consider some problems on divergence of triangular and sectoral sums for double trigonometric Fourier series. An example of a function from ∩1≤p<∞Lp with almost everywhere divergence triangular partial sums of double trigonometric Fourier series is constructed.

A remark on the divergence of strong power means of Walsh-Fourier series

AbstractF. Schipp in 1969 proved the almost everywhere p-strong summability of Walsh-Fourier series and showed that if λ(n)→∞, then there exists a function f ∈ L1[0, 1) for which the Walsh partial sums Sk(x, f) satisfy the divergence condition

BMO-Estimation and Almost Everywhere Exponential Summability of Quadratic Partial Sums of Double Fourier Series

\mathop {\lim \sup }\limits_{n \to \infty } \frac{1} {n}\sum\limits_{k = 1}^n {\left| {S_k (x,f)} \right|^{\lambda (k)} = \infty }