Tartar’s method for the Riesz–Thorin interpolation theorem

Abstract

Tartar gave an alternative proof of the Riesz–Thorin interpolation theorem for operators of strong types (1, 1) and $(\infty ,\infty )$. His method characterizes the $L^{p}$ norm in terms of the Lebesgue spaces $L^{1}$ and $L^{\infty }$, and works not only for complex Lebesgue spaces but also for real Lebesgue spaces. The aim of this paper is to extend the proof for operators of strong types $(p_{1},q_{1})$ and $(\infty ,\infty )$ with $1\le p_{1}\le q_{1}<\infty$.