Interpolation theorem for discrete net spaces

Abstract

In this paper, we study discrete net spaces np,q(M), where M is some fixed family of sets from the set of integers Z. Note that in the case when the net M is the set of all finite subsets of integers, the space np,q(M) coincides with the discrete Lorentz space lp,q(M). For these spaces, the classical interpolation theorems of Marcinkiewicz-Calderon are known. In this paper, we study the interpolation properties of discrete network spaces np,q(M),in the case when the family of sets M is the set of all finite segments from the class of integers Z, i.e. finite arithmetic progressions with a step equal to 1. These spaces are characterized by such properties that for monotonically nonincreasing sequences the norm in the space np,q(M) coincides with the norm of the discrete Lorentz space lp,q(M). At the same time, in contrast to the Lorentz spaces, the given spaces np,q(M) may contain sequences that do not tend to zero. The main result of this work is the proof of the interpolation theorem for these spaces with respect to the real interpolation method. It is shown that the scale of discrete net spaces np,q(M) is closed with respect to the real interpolation method. As a corollary, an interpolation theorem of Marcinkiewicz type is presented. These assertions make it possible to obtain strong estimates from weak estimates.