We study operators $$T:X \mapsto L_ \circ ([0,1],{\mathcal{M}},m)$$ (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space $$L_ \circ $$ are proved with the help of the Lorentz sequence spaces $$l_{p,q} $$ . Sequences of functions belonging to fixed bounded sets in the spaces $$L_{p,\infty } $$ are characterized for $$0 < p < \infty $$ and $$0 < q \leqslant p$$ . The possibility of distinguishing weak type operators (bounded in the space $$L_{p,\infty } $$ ) from operators factorizable through $$L_{p,\infty } $$ is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in $$L_{p,r} ,{\text{ }}0 < r \leqslant \infty $$ , is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.

On the basis of the Green–Lindsay initial-boundary-value problem of thermoelasticity, we formulate the corresponding variational problem in terms of displacements and temperature. Sufficient conditions of regularity of the initial data of the problem and the uniqueness of its solution are established from the energy equation of the variational problem. To prove the existence of the generalized solution (and simultaneously, as the first step to a well-justified procedure for finding its approximation), we use the method of Galerkin semidiscretization with respect to the spatial variables and show that the limit of the sequence of its approximations is the solution of the Green–Lindsay variational problem.

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter.

We propose a new method for estimating functionals in terms of higher order moduli of continuity with explicit constants. Using this method, we estimate deviations of linear methods of approximation by entire functions of finite degree and, in particular, by trigonometric polynomials. For illustration of the results, we derive estimates for the Riesz and Akhiezer–Krein–Favard averages. Bibliography: 14 titles.

It is proved that if F is an arbitrary field of characteristic p and G is a finite p-nilpotent group with a cyclic p-Sylow subgroup, then the group ring FG is serial. As a corollary, it is shown that for an arbitrary field F of characteristic 2 and any finite group G, the ring FG is serial if and only if the 2-Sylow subgroup of G is cyclic.

Inner and boundary Hölder estimates for nonnegative weak solutions of quasilinear doubly degenerate parabolic equations are established. The proof of these results is based on studing some classes B_m,1 that can be considered as extensions of the classes B₂ introduced by Ladyzhenskaya and Uraltseva and the classes B_m introduced by DiBenedetto. The embedding of the classes B_m,1 in appropriate Hölder spaces is proved. Bibliography: 20 titles.

We obtain a sufficient condition on the number of points of a Chebyshev alternance for the uniqueness of a simple partial fraction of degree n of best uniform approximation of a real-valued function on a segment of the real axis. We discuss the case where this number is greater than n + 1 and some aspects concerning approximations of constants. Bibliography: 6 titles.