Riesz convexity theorem

The logarithm, , of the least upper bound of the modulus of the bilinear form

on the set

(if or , then, respectively, , or , ) is a convex function (of a real variable) of the parameters and in the domain , if the form is real , and it is a convex function (of a real variable) in the domain , if the form is complex . This theorem was proved by M. Riesz [1].

A generalization of this theorem to linear operators is (see [3]): Let , , be the set of all complex-valued functions on some measure space that are summable to the -th power for and that are essentially bounded for . Let, further, , , , be a continuous linear operator. Then is a continuous operator from to , where

and where the norm of (as an operator from to ) satisfies the inequality (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 , , the continuity of the operator , , 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)