ABOUT SOME INTERPOLATION PROPERTIES OF FAMILIES OF FINITE DIMENSIONAL NET SPACES AND LORENZ SPACES RELATIVELY TO COMPLEX INTERPOLATION

The article provides definitions of families of finite– dimensional network spaces and Lorentz spaces and discusses some interpolation properties of families of Lorentz spaces and finite– dimensional network spaces with respect to complex interpolation. The method of complex interpolation is determined. We consider auxiliary lemmas necessary for the proof of two main interpolation theorems, with the third lemma establishing the connection between the methods of complex and real interpolation. These theorems also hold for real interpolation. The first theorem is valid for finite– dimensional Lorentz spaces, and the second is for finite– dimensional network spaces. Equality obtained in the first theorem is understood in the sense of equivalence of norms with constants independent of the choice of the number N. Equality obtained in the second theorem is understood in a similar way. In addition, the concept of an interpolation functor is considered. Earlier, interpolation properties of families of finite– dimensional network spaces and Lorentz spaces with respect to real interpolation were determined.

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