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.
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 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: Proposition (log-convexity of Lp-norms) — Each f ∈ Lp0 ∩ Lp1 satisfies: 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: Proposition — Each f in Lpθ can be written as a sum: f = g + h, where g ∈ Lp0 and h ∈ Lp1.
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.
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
Densely defined continuous operators admit unique extensions, and so we are justified in considering
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.
There are several ways to state the Riesz–Thorin interpolation theorem;[1] to be consistent with the notations in the previous section, we shall use the sumset formulation.
Indeed, the Riesz diagram of T is the collection of all points (1/p, 1/q) in the unit square [0, 1] × [0, 1] such that T is of type (p, q).
The interpolation theorem was originally stated and proved by Marcel Riesz in 1927.
[2] The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that p0 ≤ q0 and p1 ≤ q1.
Olof Thorin extended the interpolation theorem to the entire square, removing the lower-triangle restriction.
The proof of Thorin was originally published in 1938 and was subsequently expanded upon in his 1948 thesis.
Then, by Fatou’s lemma and recalling that (4) holds true for simple functions,
The proof outline presented in the above section readily generalizes to the case in which the operator T is allowed to vary analytically.
In fact, an analogous proof can be carried out to establish a bound on the entire function
The Hausdorff–Young inequality can also be established for the Fourier transform on locally compact Abelian groups.
Nevertheless, the Hilbert transform is not bounded on L1(R) or L∞(R), and so we cannot use the Riesz–Thorin interpolation theorem directly.
To see why we do not have these endpoint bounds, it suffices to compute the Hilbert transform of the simple functions 1(−1,1)(x) and 1(0,1)(x) − 1(0,1)(−x).
for all Schwartz functions f : R → C, and this identity can be used in conjunction with the Cauchy–Schwarz inequality to show that the Hilbert transform maps L2n(Rd) boundedly into itself for all n ≥ 2.
for all 2 ≤ p < ∞, and the self-adjointness of the Hilbert transform can be used to carry over these bounds to the 1 < p ≤ 2 case.
While the Riesz–Thorin interpolation theorem and its variants are powerful tools that yield a clean estimate on the interpolated operator norms, they suffer from numerous defects: some minor, some more severe.
[7] In the case of the Hilbert transform in the previous section, we were able to bypass this problem by explicitly computing the norm estimates at several midway points.
On the other hand, real interpolation methods, compared to complex interpolation methods, tend to produce worse estimates on the intermediate operator norms and do not behave as well off the diagonal in the Riesz diagram.
The off-diagonal versions of the Marcinkiewicz interpolation theorem require the formalism of Lorentz spaces and do not necessarily produce norm estimates on the Lp-spaces.
B. Mityagin extended the Riesz–Thorin theorem; this extension is formulated here in the special case of spaces of sequences with unconditional bases (cf.