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 \(1\le p_{1}\le q_{1}<\infty \).

中文翻译：

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用于 Riesz-Thorin 插值定理的 Tartar 方法

Tartar 为强类型 (1, 1) 和\((\infty ,\infty )\) 的运算符提供了 Riesz–Thorin 插值定理的替代证明。他的方法根据勒贝格空间\(L^{1}\)和\(L^{\infty }\)表征了\(L^{p}\)范数，并且不仅适用于复杂的勒贝格空间，而且也适用于真正的 Lebesgue 空间。本文的目的是扩展强类型运算符\((p_{1},q_{1})\)和\((\infty ,\infty )\)与\(1\le p_{ 1}\le q_{1}<\infty \) .