Vol 53, No 2 (2018)

The paper is devoted to special a priori estimates and Fredholm property of differential operators acting in anisotropic Sobolev spaces in ℝⁿ. Necessary conditions for a priori estimates in terms of the symbol of an operator are obtained. Under appropriate conditions imposed on the coefficients, a priori estimates are obtained in the corresponding weighted spaces.

Let S be the space of functions of regular variation and let ω = (ω₁,..., ω_n), ω_j ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B_p(ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk Uⁿ such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \) , where dm_2n(z) is the 2ndimensional Lebesgue measure on Uⁿ and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B_p(ω) and L_p(ω) (the weighted L_p-space).

In this paper, we prove uniqueness theorems and restoration formulas for coefficients of series by Vilenkin system. The series is assumed to be convergent in measure and the distribution function of the majorant of partial sums satisfies some necessary condition.

In the paper, a formula to calculate the probability that a random segment L(ω, u) in Rⁿ with a fixed direction u and length l lies entirely in the bounded convex body D ⊂ Rⁿ (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ⊂ D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ⊂ D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.