Characterization of interpolation between Grand, small or classical Lebesgue spaces

In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally G Gamma-spaces. As a direct consequence of our results any Lorentz-Zygmund space L-a,L-r (Log L)(beta), is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that 1 < a < infinity, beta not equal 0. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm. (C) 2017 Elsevier Ltd. All rights reserved.