Embedding theorems, theorems of a trace and approach for anisotropic Bαq pr (T d ) Nikol’skii-Besov spaces

Abstract

In this paper, the inequalities of different metrics and different measurements for trigonometric polynomials are obtained by using the interpolation properties of anisotropic Lorentz spaces. Further, anisotropic Nikol’skii-Besov spaces B_pr^αq(T^d) on the metric of anisotropic Lorentz spaces are defined by using the theorem on the representation. Note that in the case when d = d ₁, these spaces coincide with the classical isotropic Nikol’skii-Besov spaces, and in the case d = (1 ,..., 1) these spaces coincide with anisotropic Nikol’skii-Besov spaces, having the character of spaces with dominant mixed derivatives. In this paper the elementary embeddings are describes for these spaces with respect to all the parameters involved in their definition. By using the inequalities in different metrics the limit embedding theorem is derived for this spaces with respect to the strong parameters. An example showing unimprovability conditions on the parameters of space for save the properties of the embedding is constructed. By using the inequality of different measurements the limit theorems about the track and the continuation for the functions of these spaces are obtained. Note that the conditions in these theorems are also the best possible. In the case d = d ₁ from the received theorems follows the corresponding results, which are obtained previously in the works S.M. Nikol’skii and O.V. Besov for isotropic spaces. In the case d = (1 ,..., 1) from the received theorems follows the corresponding results, which are obtained previously in the works K.A. Bekmaganbetov and E.D. Nursultanov for anisotropic spaces with dominant mixed derivative. The obtained results show that the considered spaces have the hybrid structure of the classical isotropic spaces and anisotropic spaces with dominant mixed derivative. Namely, the function of these spaces have the same smoothness and metric properties to the variables included in one block, and different smoothness and metric properties with respect to the variables included in different blocks. The results obtained can be further used in the theory of boundary value problems of mathematical physics equations, in problems of harmonic analysis and approximation theory in anisotropic spaces.