Weight Hardy-Littlewood Inequalities for Different Powers

In this short article we obtain the non-asymptotic upper and low estimations for linear and bilinear weight Riesz's functional through the Lebesgue spaces

Central Limit Theorem and exponential tail estimations in mixed (anisotropic) Lebesgue spaces

We study the Central Limit Theorem (CLT) in the so-called mixed (anisotropic) Lebesgue-Riesz spaces and tail behavior of normed sums of centered random independent variables (vectors) with values in these spaces

Random processes and Central Limit Theorem in Besov spaces

We study sufficient conditions for the belonging of random process to certain Besov space and for the Central Limit Theorem (CLT) in these spaces. We investigate also the non-asymptotic tail behavior of normed sums of centered random independent variables (vectors) with values in these spaces. Main apparatus is the theory of mixed (anisotropic) Lebesgue-Riesz spaces, in particular so-called permutation inequality

Boundedness of Operators in Bilateral Grand Bebesgue Spaces with Exact and Weakly Exact Constant Calculation

In this article we investigate an action of some operators (not necessary to be linear or sublinear) in the so-called (Bilateral) Grand Lebesgue Spaces (GLS), in particular, double weight Fourier operators, maximal operators, imbedding operators etc. We intend to calculate an exact or at least weak exact values for correspondent imbedding constant. We obtain also interpolation theorems for GLS spaces.We construct several examples to show the exactness of offered estimations. In two last sections we introduce anisotropic Grand Lebesgue Spaces, obtain some estimates for Fourier two-weight inequalities and calculate Boyd's multidimensional indices for this spaces

A counterexample to a hypothesis of light tail of maximum distribution for continuous random processes with light finite-dimensional tails

We construct an example of a continuous centered random process with light tails of finite-dimensional distribution but with heavy tail of maximum distribution

Monte Carlo computation of multiple weak singular integrals of spherical and Volterra's type

We offer a simple method Monte Carlo for computation of Volterra's and spherical type multiple integrals with weak (integrable) singularities. An elimination of infinity of variance is achieved by incorporating singularities in the density, and we offer a highly effective way for generation of appeared multidimensional distribution. We extend offered method onto multiple Volterra's and spherical integrals with weak singularities containing parameter

Strengthening of weak convergence for Radon measures in separable Banach spaces

We prove in this short report that for arbitrary weak converging sequence of sigma-finite Borelian measures in the separable Banach space there is a compact embedded separable subspace such that this measures not only are concentrated in this subspace but weak converge therein

Fourier tranform in exponential rearrangement invariant spaces

In this article we investigate the Fourier series and transforms for the functions defined on the $[0, 2 \pi]^ d$ or $R^d$ and belonging to the exponential Orlicz and some other rearrangement invariant (r.i.) spaces.Comment: 18 pages, no figure

Quantitative lower bound for lifespan for solution of Navier-Stokes equations

We find a simple quantitative lower bound for lifespan of solution of the multidimensional initial value problem for the Navier-Stokes equations in whole space when the initial function belongs to the correspondent Lebesgue-Riesz space, and give some a priory estimations for solution in some rearrangement invariant spaces

The rate of increase for recursion with quadratic non-linearity

We investigate in this short report the rate of increase of positive numerical recursion with quadratic non-linearity. More exactly, we intent to calculate the logarithmic index of its increasing. We present also the possible application in the theory of the Navier-Stokes equations