The theorems about traces and extensions for functions from Nikol’skii-Besov spaces with generalized mixed smoothness

Abstract

The theory of embedding of spaces of differentiable functions studies important relations of differential (smoothness) properties of functions in various metrics and has wide application in the theory of boundary value problems of mathematical physics, approximation theory and other fields of mathematics.
In this article, we prove the theorems about traces and extensions for functions from Nikol’skii-Besov spaces with generalized mixed smoothness and mixed metrics. The proofs of the obtained results is based on the inequality of different dimensions for trigonometric polynomials in Lebesgue spaces with mixed metrics and the embedding theorem of classical Nikol’skii-Besov spaces in the space of continuous functions.