In mathematical analysis, 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.
Motivation First we need the following definition:
Definition. Let p0, p1 be two numbers such that 1 ≤ 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 1⁄p ↦ 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} + f 1{| f |≤1}, then we see that f 1{| f |>1} ∈ Lp0 and f 1{| 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 F {\displaystyle {\mathcal {F}}} extends to (L1 + L2) (Rd) by setting
F ( f 1 + f 2 ) = F L 1 ( f 1 ) + F L 2 ( f 2 ) {\displaystyle {\mathcal {F}}(f_{1}+f_{2})={\mathcal {F}}_{L^{1}}(f_{1})+{\mathcal {F}}_{L^{2}}(f_{2})}
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. To this end, we go back to our example and note that the Fourier transform on the sumset L1 + L2 was obtained by taking the sum of two instantiations of the same operator, namely
F L 1 : L 1 ( R d ) → L ∞ ( R d ) , {\displaystyle {\mathcal {F}}_{L^{1}}:L^{1}(\mathbf {R} ^{d})\to L^{\infty }(\mathbf {R} ^{d}),}
F L 2 : L 2 ( R d ) → L 2 ( R d ) . {\displaystyle {\mathcal {F}}_{L^{2}}:L^{2}(\mathbf {R} ^{d})\to L^{2}(\mathbf {R} ^{d}).}
These really are the same operator, in the sense that they agree on the subspace (L1 ∩ L2) (Rd). Since the intersection contains simple functions, it is dense in both L1(Rd) and L2(Rd). Densely defined continuous operators admit unique extensions, and so we are justified in considering F L 1 {\displaystyle {\mathcal {F}}_{L^{1}}} and F L 2 {\displaystyle {\mathcal {F}}_{L^{2}}} to be the same. Therefore, the problem of studying operators on the sumset Lp0 + Lp1 essentially reduces to the study of operators that map two natural domain spaces, Lp0 and Lp1, boundedly to two target spaces: Lq0 and Lq1, respectively. Since such operators map the sumset space Lp0 + Lp1 to Lq0 + Lq1, it is natural to expect that these operators map the intermediate space Lpθ to the corresponding intermediate space Lqθ.
Statement of the theorem There are several ways to state the Riesz–Thorin interpolation theorem; to be consistent with the notations in the previous section, we shall use the sumset formulation.
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