Riesz–Thorin theorem

In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.

This theorem bounds the norms of linear maps acting between L^p spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L² which is a Hilbert space, or to L¹ and L^∞. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps.

First we need the following definition:

By splitting up the function  f  in L^p_θ as the product | f | = | f |^1−θ | f |^θ and applying Hölder's inequality to its p_θ power, we obtain the following result, foundational in the study of L^p-spaces:

This result, whose name derives from the convexity of the map  ¹⁄_p ↦ log || f ||_p on [0, ∞], implies that L^p₀ ∩ L^p₁ ⊂ L^p_θ.

On the other hand, if we take the layer-cake decomposition  f  =  f 1_{{|  f  |>1}} +  f 1_{{|  f  |≤1}}, then we see that  f 1_{{|  f  |>1}} ∈ L^p₀ and  f 1_{{|  f  |≤1}} ∈ L^p₁, whence we obtain the following result:

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