Exercise 30, § 5

Let $\Phi(t,u,v)$ be a finite and continuous numerical function for $0 \leqslant t \leqslant 1$, $u \geqslant 0$, $v \geqslant 0$. For every numerical function $f \geqslant 0$ defined on $I = [0,1]$, measurable for Lebesgue measure, $f^*$ denotes the decreasing rearrangement of f (Exer. 29). In order that, for every pair of functions $f \geqslant 0$, $g \geqslant 0$, measurable (for Lebesgue measure) and bounded on I, one have

it is necessary and sufficient that the function $\Phi$ satisfy the following three conditions:

(2)

for all $t \in I$, $u \geqslant 0$, $v \geqslant 0$ and $h \geqslant 0$;

(3)

and

for all $a, h, \delta$ such that $0 \leq a \leq 1 - 2\delta ,\ h \geq 0$.

(To prove that (3) and (4) are necessary, take $f$ and $g$ to be suitable step functions; next deduce (2) from (3) and (4). To prove that the conditions are sufficient, first deduce from (2), using the continuity of $\Phi$, that

for all $t \in I,\ u \geq 0,\ v \geq 0,\ h \geq 0,\ k \geq 0$. Then using (3) and (4), prove (1) when $f$ and $g$ are step functions whose points of discontinuity in $I$ are of the form $r/n$ ($0 \leq r \leq n$), by comparing successively the integrals

and

where $f_i$ and $g_i$ differ from $f_j$ and $g_j$ only in two consecutive intervals $[ (r-1)/n, r/n ]$ and $[ r/n, (r+1)/n ]$. Finally, pass to the limit in a suitable way to prove (1) in the general case.)

If $\Phi$ is twice continuously differentiable, the conditions (2), (3) and (4) are equivalent, respectively, to

Generalize to functions $\Phi(t, u_1, \ldots, u_m)$ of any number of variables.