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". This should not be confused with somewhat different mathematical procedure of interpolationof functions. It is named after Marcel Riesz and his student G. Olof Thorin.

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.

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

:$|T|\_\{L^r\; o\; L^r\}leq\; max\; \{\; |T|\_\{L^p\; o\; L^p\},|T|\_\{L^q\; o\; L^q\}\; \}$

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"

:$r\_1=frac\{1\}\{frac\{t\}\{p\_1\}+frac\{1-t\}\{q\_1quad\; r\_2=frac\{1\}\{frac\{t\}\{p\_2\}+frac\{1-t\}\{q\_2$

"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 $L^p(Omega)$ 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 .
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*. Translated from the Russian and edited by G. P. Barker and G. Kuerti.