Riesz-Thorin theorem

In mathematics, the Riesz-Thorin theorem is a result that allows to interpolate between L^p spaces. Its usefulness stems from the fact that some of these spaces have much simpler structure than others. Usually that refers to <math>L^2<math> which is a Hilbert space, or to <math>L^1<math> and L^∞ (see examples below). 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. A related approach is to use the Marcinkiewicz theorem.

Definition

A slightly informal version of the theorem can be stated as follows:

Theorem: Assume T is a bounded linear operator from <math>L^p<math> to <math>L^p<math> and at the same time from <math>L^q<math> to <math>L^q<math>. Then it is also a bounded operator from <math>L^r<math> to <math>L^r<math> for any r between p and q.

The reason we say it is informal is because formally an operator cannot be defined on two different spaces at the same time. To formalize it we need to say: let T be a linear operator defined on a family F of functions which is dense in both <math>L^p<math> and <math>L^q<math> (for example, the family of all simple functions). And assume that Tf is in both <math>L^p<math> and <math>L^q<math> for any f in F, and that T is bounded in both norms. Then for any r between p and q we have that F is dense in <math>L^r<math>, that Tf is in <math>L^r<math> for any f in F and that T is bounded in the <math>L^r<math> norm. These three ensure that T can be extended to an operator from <math>L^r<math> to <math>L^r<math>.

In addition an inequality for the norms hold, namely

<math>||T||_{L^r\to L^r}\leq \max ||T||_{L^p\to L^p},||T||_{L^q\to L^q}<math>

A version of this theorem exists also when the domain and range of T are not identical. In this case, if T is bounded from <math>L^{p_1}<math> to <math>L^{p_2}<math> then one should draw the point <math>1/p_1, 1/p_2<math> in the unit square. The two q-s give a second point. Connect them with a straight line segment and you get the r-s for which T is bounded. Here is again the almost formal version

Theorem: Assume T is a bounded linear operator from <math>L^{p_1}<math> to <math>L^{p_2}<math> and at the same time from <math>L^{q_1}<math> to <math>L^{q_2}<math>. Then it is also a bounded operator from <math>L^{r_1}<math> to <math>L^{r_2}<math> where

<math>r_1=\frac{1}{\frac{t}{p_1}+\frac{1-t}{q_1}}\quad r_2=\frac{1}{\frac{t}{p_2}+\frac{1-t}{q_2}}<math>

and t is any number between 0 and 1.

The perfect formalization is done as in the simpler case.

One last generalization is that the theorem holds for <math>L^p(\Omega)<math> for any measure space Ω. In particular it holds for the <math>l^p<math> spaces.

Application examples

The first example is the Fourier operator, namely let T be the operator that takes a function on the unit circle and outputs its Fourier series. Parseval's theorem shows that T is bounded from <math>L^2<math> to <math>l^2<math> with norm 1. On the other hand, clearly,

<math>|(Tf)(n)|=|\widehat{f}(n)|=\left|\int_0^{2\pi}f(t)e^{int}\,dt\right|\leq\int_0^{2\pi}|f(t)|\,dt<math>

so T is bounded from <math>L^1<math> to l^∞ with norm 1. Therefore we may invoke the Riesz-Thorin theorem to get, for any 1 < p < 2 that T, as an operator from <math>L^p<math> to <math>l^q<math> with norm 1, where

<math>\frac{1}{p}+\frac{1}{q}=1.<math>

In a short formula, this says that

<math>\left(\sum_{n=-\infty}^{\infty}|\widehat{f}(n)|^q\right)^{1/q}\leq

\left(\int_0^{2\pi}|f(t)|^p\,dt\right)^{1/p}.<math>

This is the well known Hausdorff-Young inequality. It might be interesting to note that for p > 2 the natural extrapolation of this inequality fails, and the fact that a function belongs to <math>L^p<math>, does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in <math>l^2<math>.

Convolution operators

Let f be an integrable function and let T be the operator of convolution with f, i.e.

<math>\,Tg = f * g.<math>

It is well known that T is bounded from <math>L^1<math> to <math>L^1<math> and it is trivial that it is bounded from L^∞ to L^∞ (both bounds are by <math>||f||_1<math>). Therefore the Riesz-Thorin theorem gives

<math>||f*g||_p\leq ||f||_1||g||_p.<math>

We take this inequality and switch the role of the operator and the operand, or in other words, we think of S as the operator of convolution with g, and get that S is bounded from <math>L^1<math> to <math>L^p<math>. Further, since g is in <math>L^p<math> we get, for the same trivial reason as above, that S is bounded from <math>L^q<math> to L^∞, where again <math>1/p+1/q=1<math>. So interpolating we get

<math>||f*g||_s\leq ||f||_r||g||_p<math>

where the connection between p, r and s is

<math>\frac{1}{r}+\frac{1}{p}=1+\frac{1}{s}.<math>

Thorin's contribution

Riesz and Thorin did not publish this result together nor concurrently. The original proof of Riesz was a long and difficult calculation. Thorin discovered a far more elegant proof, which we will now sketch very briefly. He defined a generalization of the <math>L^p<math> spaces to complex p. After defining an appropriate complex function, it turned out that the boundedness of T on <math>L^p<math> gave that this complex function was bounded on the line <math>p+iy<math>. The boundedness on <math>L^q<math> gave that the function was bounded on the line <math>q+iy<math>. Applying the Phragmén-Lindelöf principle (a kind of maximum principle for infinite domains) one gets that the function is bounded on the entire strip between these two lines, and in particular in the point r.

Attempts at generalizing this approach were largely successful, and led to the notion of complex interpolation. In rough terms, we say that two Banach spaces have complex interpolation between them if a similar procedure can be applied to get the boundedness of an operator T on a continuum of Banach spaces "between them". See for example Sobolev space.