Riesz convexity theorem

The logarithm, $\mathop{\rm ln} M( \alpha , \beta )$, of the least upper bound of the modulus $M( \alpha , \beta )$ of the bilinear form

$$\sum_{i=1} ^ { m } \sum_{j=1} ^ { n } a _ {ij} x _ {i} y _ {j}$$

on the set

$$\sum_{i=1} ^ { m } | x _ {i} | ^ {1/ \alpha } \leq 1,\ \ \sum_{j=1} ^ { m } | y _ {j} | ^ {1/ \beta } \leq 1$$

(if $\alpha = 0$ or $\beta = 0$, then, respectively, $| x _ {i} | \leq 1$, $i = 1 \dots m$ or $| y _ {j} | \leq 1$, $j = 1 \dots n$) is a convex function (of a real variable) of the parameters $\alpha$ and $\beta$ in the domain $\alpha \geq 0$, $\beta \geq 0$ if the form is real $( a _ {ij} , x _ {i} , y _ {j} \in \mathbf R _ {+} )$, and it is a convex function (of a real variable) in the domain $0 \leq \alpha , \beta \leq 1$, $\alpha + \beta \geq 1$ if the form is complex $( a _ {ij} , x _ {i} , y _ {j} \in \mathbf C )$. This theorem was proved by M. Riesz [1].

A generalization of this theorem to linear operators is (see [3]): Let $L _ {p}$, $1 \leq p \leq \infty$, be the set of all complex-valued functions on some measure space that are summable to the $p$- th power for $1 \leq p < \infty$ and that are essentially bounded for $p = \infty$. Let, further, $T: L _ {p _ {i} } \rightarrow L _ {q _ {i} }$, $1 \leq p _ {i} , q _ {j} \leq \infty$, $i = 0, 1$, be a continuous linear operator. Then $T$ is a continuous operator from $L _ {p _ {t} }$ to $L _ {q _ {t} }$, where

$$\frac{1}{p _ {t} } = 1- \frac{t}{p _ {0} } + \frac{t}{p _ {1} } ,\ \ \frac{1}{q _ {t} } = 1- \frac{t}{q _ {0} } + \frac{t}{q _ {1} } ,\ \ t \in [ 0, 1],$$

and where the norm $k _ {t}$ of $T$( as an operator from $L _ {p _ {t} }$ to $L _ {q _ {t} }$) satisfies the inequality $k _ {t} \leq k _ {0} ^ {1-} t k _ {1} ^ {t}$( i.e. it is a logarithmically convex function). This theorem is called the Riesz–Thorin interpolation theorem, and sometimes also the Riesz convexity theorem [4].

The Riesz convexity theorem is at the origin of a whole trend of analysis in which one studies interpolation properties of linear operators. Among the first generalizations of the Riesz convexity theorem is the Marcinkiewicz interpolation theorem [5], which ensures for $1 \leq p _ {i} \leq q _ {i} \leq \infty$, $i = 0, 1$, the continuity of the operator $T: L _ {p _ {t} } \rightarrow L _ {q _ {t} }$, $t \in ( 0, 1)$, under weaker assumptions than those of the Riesz–Thorin theorem. See also Interpolation of operators.

References

[1]	M. Riesz, "Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires" Acta Math. , 49 (1926) pp. 465–497
[2]	G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[3]	G.O. Thorin, "An extension of a convexity theorem due to M. Riesz" K. Fysiogr. Saallskap. i Lund Forh. , 8 : 14 (1936)
[4]	E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)
[5]	J. Marcinkiewicz, "Sur l'interpolation d'opérateurs" C.R. Acad. Sci. Paris , 208 (1939) pp. 1272–1273
[6]	S.K. Krein, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)
[7]	H. Triebel, "Interpolation theory" , Springer (1978)