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 lp space| spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to which is a Hilbert space, or to and. 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:
By splitting up the function in as the product and applying Hölder's inequality to its power, we obtain the following result, foundational in the study of -spaces:
This result, whose name derives from the convexity of the map on, implies that.
On the other hand, if we take the layer-cake decomposition, then we see that and, whence we obtain the following result:
In particular, the above result implies that is included in, the sumset of and 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. For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps boundedly into, and Plancherel's theorem shows that the Fourier transform maps boundedly into itself, hence the Fourier transform extends to by setting
for all and. It is therefore natural to investigate the behavior of such operators on the intermediate subspaces.
To this end, we go back to our example and note that the Fourier transform on the sumset was obtained by taking the sum of two instantiations of the same operator, namely
These really are the same operator, in the sense that they agree on the subspace. Since the intersection contains simple functions, it is dense in both and. Densely defined continuous operators admit unique extensions, and so we are justified in considering and to be the same.
Therefore, the problem of studying operators on the sumset essentially reduces to the study of operators that map two natural domain spaces, and, boundedly to two target spaces: and, respectively. Since such operators map the sumset space to, it is natural to expect that these operators map the intermediate space to the corresponding intermediate space.

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.
In other words, if is simultaneously of type and of type, then is of type for all. In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, the Riesz diagram of is the collection of all points in the unit square such that is of type. The interpolation theorem states that the Riesz diagram of is a convex set: given two points in the Riesz diagram, the line segment that connects them will also be in the diagram.
The interpolation theorem was originally stated and proved by Marcel Riesz in 1927. The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that and. 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.

Sketch of proof

The proof of the Riesz–Thorin interpolation theorem relies crucially on the Hadamard three-lines theorem to establish the requisite bounds. By the characterization of the dual spaces of -spaces, we see that
By suitably defining variants and of and for each in, we obtain the entire function
whose value at is
We can then use the hypotheses to establish upper bounds of on the lines and, whence the Hadamard three-lines theorem establishes the interpolated bound of on the line. It now suffices to check that the bound at is what we wanted.

Interpolation of analytic families of operators

The proof outline presented in the above section readily generalizes to the case in which the operator is allowed to vary analytically. In fact, an analogous proof can be carried out to establish a bound on the entire function
from which we obtain the following theorem of Elias Stein, published in his 1956 thesis:
The theory of real Hardy spaces and the space of bounded mean oscillations permits us to wield the Stein interpolation theorem argument in dealing with operators on the Hardy space and the space of bounded mean oscillations; this is a result of Charles Fefferman and Elias Stein.

Applications

Hausdorff–Young inequality

It has been shown in the [|first section] that the Fourier transform maps boundedly into and into itself. A similar argument shows that the Fourier series operator, which transforms periodic functions into functions whose values are the Fourier coefficients
maps boundedly into and into. The Riesz–Thorin interpolation theorem now implies the following:
where and. This is the Hausdorff–Young inequality.
The Hausdorff–Young inequality can also be established for the Fourier transform on locally compact Abelian groups. The norm estimate of 1 is not optimal. See the main article for references.

Convolution operators

Let be a fixed integrable function and let be the operator of convolution with, i.e., for each function we have.
It is well known that is bounded from to and it is trivial that it is bounded from to . Therefore the Riesz–Thorin theorem gives
We take this inequality and switch the role of the operator and the operand, or in other words, we think of as the operator of convolution with, and get that is bounded from to Lp. Further, since is in we get, in view of Hölder's inequality, that is bounded from to, where again. So interpolating we get
where the connection between p, r and s is

The Hilbert transform

The Hilbert transform of is given by
where p.v. indicates the Cauchy principal value of the integral. The Hilbert transform is a Fourier multiplier operator with a particularly simple multiplier:
It follows from the Plancherel theorem that the Hilbert transform maps boundedly into itself.
Nevertheless, the Hilbert transform is not bounded on or, 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 and. We can show, however, that
for all Schwartz functions, and this identity can be used in conjunction with the Cauchy–Schwarz inequality to show that the Hilbert transform maps boundedly into itself for all. Interpolation now establishes the bound
for all, and the self-adjointness of the Hilbert transform can be used to carry over these bounds to the case.

Comparison with the real interpolation method

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. Note first that the complex-analytic nature of the proof of the Riesz–Thorin interpolation theorem forces the scalar field to be. For extended-real-valued functions, this restriction can be bypassed by redefining the function to be finite everywhere—possible, as every integrable function must be finite almost everywhere. A more serious disadvantage is that, in practice, many operators, such as the Hardy–Littlewood maximal operator and the Calderón–Zygmund operators, do not have good endpoint estimates. 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. This is cumbersome and is often not possible in more general scenarios. Since many such operators satisfy the weak-type estimates
real interpolation theorems such as the Marcinkiewicz interpolation theorem are better-suited for them. Furthermore, a good number of important operators, such as the Hardy-Littlewood maximal operator, are only sublinear. This is not a hindrance to applying real interpolation methods, but complex interpolation methods are ill-equipped to handle non-linear operators. 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 -spaces.

Mityagin's theorem

B. Mityagin extended the Riesz–Thorin theorem; this extension is formulated here in the special case of spaces of sequences with unconditional bases.
Assume:
Then
for any unconditional Banach space of sequences, that is, for any and any,.
The proof is based on the Krein–Milman theorem.