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 Hunt's inequality.     

On unboundedness of maximal operators for directional Hilbert transforms

We show that for any infinite set of unit vectors in the maximal operator defined by

is not bounded in .

     

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.     

On classes of everywhere divergent power series

We prove the everywhere divergence of series $$ \sum_{n=0}^\infty a_n e^{i\rho_n}e^{inx}, \quad\text{and}\quad \sum_{n=0}^\infty {(-1)}^{[\rho_n]}a_n \cos nx, $$ for sequences a _n and ρ _n satisfying some extremal conditions. These results generalize some well known examples of everywhere divergent power and trigonometric series.     

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 G _δσ type set of zero measure.     

ChemInform Abstract: On Compound Formation of M′O:M2O3. Part 11. Crystal Structure of BaIn2O4.

Single crystals of the title compound, obtained from a melt after reacting intimate mixtures of BaCO₃ and In₂O₃ at 840 °C (12 h), belong to the monoclinic space group P2₁/a with Z=16.     

ChemInform Abstract: Boron Occupied Condensed Clusters in Ta5Ge3By (y ≈ 0.7).

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