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Concept
Riesz–Thorin theorem
Summary
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 Lp 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:
Definition. Let p0, p1 be two numbers such that 0 < p0 < p1 ≤ ∞. Then for 0 < θ < 1 define pθ by: 1/pθ = 1 − θ/p0 + θ/p1.
By splitting up the function f in Lpθ as the product f = f 1−θ f θ and applying Hölder's inequality to its pθ power, we obtain the following result, foundational in the study of Lp-spaces:
This result, whose name derives from the convexity of the map ↦ log f p on [0, ∞], implies that Lp0 ∩ Lp1 ⊂ Lpθ.
On the other hand, if we take the layer-cake decomposition f = f 1 + f 1, then we see that f 1 ∈ Lp0 and f 1 ∈ Lp1, whence we obtain the following result:
In particular, the above result implies that Lpθ is included in Lp0 + Lp1, the sumset of Lp0 and Lp1 in the space of all measurable functions. Therefore, we have the following chain of inclusions:
In practice, we often encounter operators defined on the sumset Lp0 + Lp1. For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps L1(Rd) boundedly into L∞(Rd), and Plancherel's theorem shows that the Fourier transform maps L2(Rd) boundedly into itself, hence the Fourier transform extends to (L1 + L2) (Rd) by setting
for all f1 ∈ L1(Rd) and f2 ∈ L2(Rd). It is therefore natural to investigate the behavior of such operators on the intermediate subspaces Lpθ.
Official source
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