Riesz-Thorin theorem

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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. This should not be confused with somewhat different mathematical procedure of interpolation of functions.

This theorem deals with linear maps acting between L^p spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to $L 2$ which is a Hilbert space, or to $L 1$ 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.

1 Definition
2 Application examples
- 2.1 Hausdorff−Young inequality
- 2.2 Convolution operators
3 Thorin's contribution
4 Mityagin's theorem
5 References

[edit] Definition

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

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

This is informal because an operator cannot formally 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 $L p$ and $L q$ (for example, the family of all simple functions). And assume that Tf is in both $L p$ and $L q$ 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 $L r$ , that Tf is in $L r$ for any f in F and that T is bounded in the $L r$ norm. These three ensure that T can be extended to an operator from $L r$ to $L r$ .

In addition an inequality for the norms holds, namely

$\|T\|_{L^r\to L^r}\leq \max \{ \|T\|_{L^p\to L^p},\|T\|_{L^q\to L^q} \}$

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 $L^{p_1}$ to $L^{p_2}$ then one should draw the point $1 / p 1,1 / p 2$ 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 $L^{p_1}$ to $L^{p_2}$ and at the same time from $L^{q_1}$ to $L^{q_2}$ . Then it is also a bounded operator from $L^{r_1}$ to $L^{r_2}$ where

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

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 $L p (Ω)$ for any measure space Ω. In particular it holds for the $\ell^p$ spaces.

[edit] Application examples

[edit] Hausdorff−Young inequality

We consider the Fourier operator, namely let T be the operator that takes a function $f$ on the unit circle and outputs the sequence of its Fourier coefficients $\widehat{f}(n)=\frac{1}{2\pi}\int_0^{2\pi}e^{-inx}f(x)\,dx, n=0,\pm1,\pm2,....$ . Parseval's theorem shows that T is bounded from $L 2$ to $\ell^2$ with norm 1. On the other hand, clearly,

$|(Tf)(n)|=|\widehat{f}(n)|=\left|\frac{1}{2\pi}\int_0^{2\pi}e^{-int}f(t)\,dt\right|\leq \frac{1}{2\pi} \int_0^{2\pi}|f(t)|\,dt$

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

$\frac{1}{p}+\frac{1}{q}=1.$

In a short formula, this says that

$\left(\sum_{n=-\infty}^{\infty}|\widehat{f}(n)|^q\right)^{1/q}\leq \left( \frac{1}{2\pi}\int_0^{2\pi}|f(t)|^p\,dt\right)^{1/p}.$

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 $L p$ , does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in $\ell^2$ .

[edit] Convolution operators

Let f be a fixed integrable function and let T be the operator of convolution with f, i.e., for each function g we have

$\,Tg = f * g.$

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

$\|f*g\|_p\leq \|f\|_1\|g\|_p.$

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 $L 1$ to $L p$ . Further, since g is in $L p$ we get, in view of Hölder's inequality, that S is bounded from $L q$ to L^∞, where again $1 / p + 1 / q = 1$ . So interpolating we get

$\|f*g\|_s\leq \|f\|_r\|g\|_p$

where the connection between p, r and s is

$\frac{1}{r}+\frac{1}{p}=1+\frac{1}{s}.$

[edit] Thorin's contribution

The original proof of this theorem, published in 1926 by Marcel Riesz, was a long and difficult calculation. Riesz' student G. Olof Thorin subsequently discovered a far more elegant proof and published it in 1939. The English mathematician J. E. Littlewood once enthusiastically referred to Thorin's proof as "the most impudent idea in mathematics".

Here is a brief sketch of that proof:

One of its main ingredients is the following rather well known result about analytic functions. Suppose that $F (z)$ is a bounded analytic function on the two lines $1 / p + i y$ and $1 / q + i y$ and on the strip between these two lines. Suppose also that $|F(z)|\le 1$ at every point $z$ on those two lines. Then, by applying the Phragmén-Lindelöf principle (a kind of maximum principle for infinite domains) one gets that $|F(z)|\le 1$ at every point between these two lines, and in particular at the point $z = 1 / r$ .

Thorin ingeniously defined a special analytic function $F$ connected with the operator $T$ . He used the fact that $T$ is bounded on $L p$ to deduce that $|F(z)|\le 1$ on the line $1 / p + i y$ , and, analogously, he used the boundedness of $T$ on $L q$ to deduce that $|F(z)|\le 1$ on the line $1 / q + i y$ . Then, after using the result mentioned above to give that $|F(1/r)|\le 1$ , he was able to show that this implies that $T$ is bounded on $L r$ .

Thorin obtained this function $F$ with the help of a generalized notion of an analytic function whose values are elements of $L p$ spaces instead of being complex numbers. In the 1960's Alberto Calderón adapted and generalized Thorin's ideas to develop the method of complex interpolation. Suppose that $A 0$ and $A 1$ are two Banach spaces which are continuously contained in some suitable larger space. Calderon's method enables one to construct a family of new Banach spaces $A t$ , for each $t$ with $0 < t < 1$ which are ``between" $A 0$ and $A 1$ and have the ``interpolation" property that every linear operator $T$ which is bounded on $A 0$ and on $A 1$ is also bounded on each of the complex interpolation spaces $A t$ .

Calderon's spaces have many applications. See for example Sobolev space.

[edit] Mityagin's theorem

B.Mityagin extended the Riesz-Thorin theorem; we formulate the extension in the special case of spaces of sequences with unconditional bases (cf. below).

Assume $\|A\|_{\ell_1 \to \ell_1} \leq M$ , $\|A\|_{\ell_\infty \to \ell_\infty} \leq M$ . Then $\|A\|_{X \to X} \leq M$ for any unconditional Banach space of sequences $X$ (that is, for any $(x_i) \in X$ and any $(\epsilon_i) \in \{ -1, +1 \}^\infty$ , $\| (\epsilon_i x_i) \|_X = \| (x_i) \|_X$ ).

The proof is based on the Krein-Milman theorem.

[edit] References

B.S.Mitjagin [Mityagin], An interpolation theorem for modular spaces (Russian), Mat. Sb. (N.S.) 66 (108) 1965 473–482
I.M.Glazman, Yu.I.Lyubich, Finite-dimensional linear analysis: a systematic presentation in problem form, Translated from the Russian and edited by G. P. Barker and G. Kuerti. The M.I.T. Press, Cambridge, Mass.-London, 1974. xvi+520 pp.

Riesz-Thorin theorem

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Application examples

[edit] Hausdorff−Young inequality

[edit] Convolution operators

[edit] Thorin's contribution

[edit] Mityagin's theorem

[edit] References

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