Interpolation properties of anisotropic Bq pr (Td) Nikol’skii-Besov spaces and embedding theorems

Abstract

In this paper we study interpolation properties of anisotropic spaces of Nikol’skii-Besov with respect to the anisotropic interpolation A θq , where 0 < θ = (θ 1,..., θ n) < 1, 1 ≤ q = (q 1,..., q n) ≤ ∞. It shows that in the case −∞ < α = (α 1,..., α n) < ∞, 1 < p = (p 1,..., p n) < ∞ and 1 ≤ q = (q 1,..., q n), r = (r 1,..., r n) ≤ ∞, d = (d 1,..., d n) the Nikol’skii-Besov spaces B pr αq (T d ) are the retract spaces l q α (L pr (T d )), and corresponding interpolation theorem follows from the interpolation properties of anisotropic l q α (A) spaces. The interpolation properties of anisotropic Sobolev spaces W pr α (T d ) with dominant mixed derivative are described as a corollary. In the second part of the work the Nikol’skii inequality of different metrics for trigonometric polinomials with the spectrum from parallelepipeds in the anisotropic Lorentz spaces L qτ (T d ) is proved. On the basis of the inequality and interpolation theorems the embedding theorems for anisotropic Nikol’skii-Besov spaces B pr ατ (T d ) and anisotropic Lorentz spaces L qτ (T d ) are obtained. The relations, which connect the spaces parameters α, p and q, i.e. α = (1/p − 1/q) are the limiting. This relations also can not be improved. The obtained theorems generalize the corresponding results from the works of K.A. Bekmaganbetov and E.D. Nursultanov for the case d = (1,..., 1).