Sufficient conditions for the precompactness of sets in Local Morrey-type spaces

Abstract

In this paper we give sufficient conations for the pre-compactness of sets іп local Morrey-type spaces LMpg>w(.) (Rn). For w(r) = r-X, Ө = to, 0 < Л < ^ there follows a known result for the Morrey spaces Mp(Rn). In the case Л = 0 this is the well-known Frechet-Kolmogorov theorem. The pre-compactness of sets іп Morrey spaces was investigated іп the works [1, 2], and іп generalized Morrey spaces Мр“( )(Кп) іп the works [3, 4]. The aim of this paper is to generalize these results to the case of Local Morrey-type spaces LMpe,w(.) (Rn). 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 supfeS || f (• + u) — f G)||LM e =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 sup А8f — f ||L (B(0 r)\B(0 r)) = 0.