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 L^p 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 to non-linear maps.

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 that is dense in both L^p and L^q (for example, the family of all simple functions). And assume that Tƒ is in both L^p and L^q for any ƒ 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 Tƒ is in L^r for any ƒ 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 / p₁,1 / p₂) 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 $L^{r_2}$ with

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

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.

Convexity

Another more general form of the theorem is as follows (Dunford & Schwartz 1958, §VI.10.11). Suppose that μ₁ and μ₂ are two measures on possibly different measure spaces. Let T be a linear mapping from the space of μ₁-integrable functions into the space of μ₂-measurable functions, and for 1 ≤ p,q ≤ ∞, define $\scriptstyle{|T|_{p,q}}$ to be the operator norm of a continuous extension of T to

$T:L^p(\mu_1)\to L^q(\mu_2)$

if such an extension exists, and ∞ otherwise. Then the theorem asserts that the function

f(a,b) = log  | T | _{1 / a,1 / b}

is convex in the rectangle (a,b) ∈ [0,1]×[0,1].

Application examples

Hausdorff−Young inequality

Main article: 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

$\widehat{f}(n)=\frac{1}{2\pi}\int_0^{2\pi}e^{-inx}f(x)\,dx, n=0,\pm1,\pm2,\dots.$

Parseval's theorem shows that T is bounded from L² 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 L¹ 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 Hausdorff–Young inequality.

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 L¹ to L¹ 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 L¹ 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

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

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 + iy and 1 / q + iy 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 + iy, and, analogously, he used the boundedness of T on L^q to deduce that $|F(z)|\le 1$ on the line 1 / q + iy. 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 1960s Alberto Calderón adapted and generalized Thorin's ideas to develop the method of complex interpolation. Suppose that A₀ and A₁ 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<sub>0</sub> and A<sub>1</sub> and have the ``interpolation" property that every linear operator T which is bounded on A₀ and on A₁ is also bounded on each of the complex interpolation spaces A_t.

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

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.

References

• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Parts I and II, Wiley-Interscience .
• Glazman, I.M.; Lyubich, Yu.I. (1974), Finite-dimensional linear analysis: a systematic presentation in problem form, Cambridge, Mass.: The M.I.T. Press . Translated from the Russian and edited by G. P. Barker and G. Kuerti.
• Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, ISBN 3-540-12104-8, MR0717035 .
• Mitjagin [Mityagin], B.S. (1965), "An interpolation theorem for modular spaces (Russian)", Mat. Sb. (N.S.) 66 (108): 473–482 .
• Thorin, G. O. (1948), "Convexity theorems generalizing those of M. Riesz and Hadamard with some applications", Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 9: 1–58, MR0025529