Characterization of interpolation between Grand, small or classical Lebesgue spaces

Abstract

In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz–Zygmund spaces or more generally GΓ-spaces. As a direct consequence of our results any Lorentz–Zygmund space La,r(LogL)β, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that 1<a<∞,β≠0. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm.