Characterization of interpolation between Grand, small or classical Lebesgue spaces

TL;DR

This paper characterizes the interpolation spaces between Grand, small, and classical Lebesgue spaces, showing they are Lorentz-Zygmund or $G ext{-} ext{Gamma}$-spaces, with implications for understanding their structure and relationships.

Contribution

It provides a detailed description of the interpolation spaces between various Lebesgue spaces, identifying them as Lorentz-Zygmund or $G ext{-} ext{Gamma}$-spaces, and computes their K-functional.

Findings

Lorentz-Zygmund spaces are interpolation spaces between Grand and small Lebesgue spaces.

The K-functional for these interpolation spaces is explicitly computed.

Any Lorentz-Zygmund space $L^{a,r}( ext{Log} ext{L})^eta$ is an interpolation space between two Grand or small 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 $G\Gamma$-spaces. As a direct consequence of our results any Lorentz-Zygmund space $L^{a,r}({\rm Log}\, L)^\beta$, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that $ 1<a<\infty, \beta \not= 0$. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm.