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

Abstract

In this paper we give sufficient conditions for the precompactness of sets in general global Morrey-type spaces GMpθ,w(·)(ℝn). In the case θ = ∞, it follows the result for the generalized Morrey spaces Mpw(⋅), and for w(r) = r−λ, θ = ∞, 0≤λ≤np there follows the known result for the Morrey spaces Mpλ(ℝn). In the case λ = 0 this is the well-known Frechet-Kolmogorov theorem. The pre-compactness of sets in Morrey spaces was investigated in many works, and in generalized Morrey spaces Mpw(⋅)(ℝn) in our previous works. The aim of this paper is to generalize these results to the case of general global Morrey-type spaces GMpθ,w(·)(ℝn).