Riesz convexity theorem

The logarithm, $\ln M(\alpha ,\beta )$, 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

$$\sum _{i=1}^m|x_i{|}^{1/\alpha }\le 1,\sum _{j=1}^m|y_j{|}^{1/\beta }\le 1$$

(if $\alpha =0$ or $\beta =0$, then, respectively, $|x_i|\le 1$, $i=1\dots m$ or $|y_j|\le 1$, $j=1\dots n$) is a convex function (of a real variable) of the parameters $\alpha$ and $\beta$ in the domain $\alpha \ge 0$, $\beta \ge 0$ 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 $0\le \alpha ,\beta \le 1$, $\alpha +\beta \ge 1$ 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 $L_p$, $1\le p\le \mathrm{\infty }$, be the set of all complex-valued functions on some measure space that are summable to the $p$- th power for $1\le p<\mathrm{\infty }$ and that are essentially bounded for $p=\mathrm{\infty }$. Let, further, $T:L_{p_i}\to L_{q_i}$, $1\le p_i,q_j\le \mathrm{\infty }$, $i=0,1$, 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 $k_t\le k_0^{1-}t{k}_1^t$( 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 $1\le p_i\le q_i\le \mathrm{\infty }$, $i=0,1$, the continuity of the operator $T:L_{p_t}\to L_{q_t}$, $t\in (0,1)$, under weaker assumptions than those of the Riesz–Thorin theorem. See also Interpolation of operators.

References[edit]