Sufficient conditions for pre-compactness of sets in the generalized Morrey spaces

Abstract

In this paper, we present suficient conditions for pre - compactness of sets in the generalized Morrey spaces M_p^w(·)(Rⁿ). From this theorem for the case of w(r) = r^-λ ; 0≤λ≤ n/p follows the well - known result for the Morrey space M_p^λ (Rⁿ) and in the case λ = 0 this is the well - known Frechet-Kolmogorov theorem. Pre proved some lemmas on the estimation of the average in the generalized Morrey space. These lemmas gives independent interest. It is discussed the necessity of the obtained conditions.