Abstract
Sufficient conditions for the precompactness of sets in Local Morrey-type spacesIn this paper we give sufficient conditions for the pre-compactness of sets in local Morrey-type spacesp there follows a known result for the Morrey spaces M λ p (R n ).In the case λ = 0 this is the well-known Frechet-Kolmogorov theorem.The pre-compactness of sets in Morrey spaces was investigated in the works [1, 2], and in generalized Morrey spaces M w(•) p (R n ) in the works [3, 4].The aim of this paper is to generalize these results to the case of Local Morrey-type spaces LM pθ,w(•) (R n ).By using theorem of pre-compactness set in local Morrey-type spaces, compact of operators can be checked in this spaces, since compact operator transfers from bounded set of one space to pre-compact set of another space.In this paper, the conditions of precompactness of sets in local spaces of Morrey type are given in terms of the difference of the function limu→0 sup f ∈S f (• + u) -f (•) LM pθ,w = 0. Earlier, the necessary and sufficient conditions for precompactness of sets in local spaces of Morrey type were published in [5], which were given in terms of the mean functions lim