Abstract The present paper continues the series of the papers [Amaglobeli and Bokelavadze, Exponential groups. III. Groups which are exact for tensor complement, Herald of Omsk University, 2009, Myasnikov and Remeslennikov, Sibirsk. Mat. Zh. 35: 1106–1118, 1994, Myasnikov and Remeslennikov, Internat. J. Algebra Comput. 6: 687–711, 1996] and is dedicated to the construction of basic principles of the theory of power group varieties and tensor completions of groups in a variety. We study the relationship between free groups of a given variety for various rings of scalars. Varieties of abelian power groups are described.

Abstract In this paper, we estimate the growth order of entire solutions of some algebraic differential equations and improve the related result of Bergweiler, Barsegian and others. We also give some examples to show that our result is sharp in special cases.

Abstract The sufficient conditions of well-posedness of the weighted Cauchy problem for higher order linear functional differential equations with deviating arguments, whose coefficients have nonintegrable singularities at the initial point, are found.

Abstract Using the Bellman function method, we prove that a Haar wavelet system of rank N (N ∈ ℕ, N ≥ 2) is an unconditional basis in , 1 < p < ∞, if and only if .

Abstract Refined direct and converse theorems of trigonometric approximation are proved in the variable exponent Lebesgue spaces with weights satisfying some Muckenhoupt Ap -condition. As a consequence, the refined versions of Marchaud and its converse inequalities are obtained.

Abstract The boundedness of maximal operators on the weighted Choquet space and the Choquet–Morrey space is established. These results are used to study Carleson embeddings for weighted Sobolev spaces.

Abstract The interior and exterior Robin problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so-called “superformula” introduced by J. Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained. The computed results are found to be in good agreement with the theoretical findings on Fourier series expansion presented by L. Carleson.

Abstract First we compute the trace space of Besov spaces – characterized via atomic decompositions – on fractals Γ, for parameters 0 < p < ∞, 0 < q ≤ min(1, p) and s = (n – d)/p. New Besov spaces on fractals are defined via traces for 0 < p, q ≤ ∞, s ≥ (n – d)/p and some embedding assertions are established. We conclude by studying the compactness of the trace operator TrΓ by giving sharp estimates for entropy and approximation numbers of compact embeddings between Besov spaces. Our results on Besov spaces remain valid considering the classical spaces defined via differences. The trace results are used to study traces in Triebel–Lizorkin spaces as well.

Abstract The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.

Abstract Let 1 ≤ q < p < ∞, and ℝ be the real line. Hörmander showed that any bounded linear translation invariant operator from Lp (ℝ) to Lq (ℝ) is trivial. Blozinski obtained an analogy to Hörmander in Lorentz spaces on the real line. In this paper, we generalize Blozinski's result in Lorentz–Zygmund spaces. Also, Bochkarev proved an inequality related to the Hausdorff–Young–Riesz theorem in Lorentz spaces, and the sharpness of the inequality. We improve Bochkarev's inequality in Lorentz–Zygmund spaces, and prove the sharpness of our inequality.