Exercise 14, § 6

Integration, IV, p. 137

Bourbaki marks this one as harder than the rest.

Let $p$ and $q$ be any two finite real numbers $> 0$ and let $f(x_1, x_2, \ldots, x_n)$ be a continuous numerical function defined on $\mathbf{R}^n$ .

a) Let $\mu$ be Lebesgue measure on $X = [0, 1]$ . In order that, for every system of $n$ functions $g_k \in \mathscr{L}^p$ , the function $f(g_1, g_2, \ldots, g_n)$ belong to $\mathscr{L}^q$ , it is necessary and sufficient that there exist a number $a > 0$ such that

|f(x_1, \ldots, x_n)|^q \leqslant a (1 + |x_1| + \ldots + |x_n|)^p.

(To see that the condition is necessary, argue by contradiction, supposing that for every integer $m > 0$ , there exists a point $(x_{1m}, \ldots, x_{nm})$ of $\mathbf{R}^n$ such that

|f(x_{1m}, \ldots, x_{nm})|^q \geq m(1 + |x_{1m}| + \ldots + |x_{nm}|)^p .

Show that there would then exist in X a sequence $(A_m)$ of pairwise disjoint intervals such that, on setting $g_k(t) = x_{km}$ for every $t \in A_m$ and $g_k(t) = 0$ for every point t that does not belong to any of the $A_m$ , each of the functions $g_k$ would belong to $\mathscr{L}^p$ but $f(g_1, \ldots, g_n)$ would not belong to $\mathscr{L}^q$ .)

b) Let $\mu$ be Lebesgue measure on $\mathbf{R}$ . In order that, for every system of n functions $g_k \in \mathscr{L}^p$ , the function $f(g_1, \ldots, g_n)$ belong to $\mathscr{L}^q$ , it is necessary and sufficient that there exist a number $b > 0$ such that

|f(x_1, \ldots, x_n)|^q \leq b(|x_1| + |x_2| + \ldots + |x_n|)^p

(same method).

Solution

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