Exercise 25, § 5

Integration, V, p. 105

Bourbaki marks this one as harder than the rest.

a) If $u$ and $v$ are any two real numbers, prove, for $1 < p < +\infty$ , the inequality

|u + v|^p \leq |u|^p + pvu|u|^{p-2} + a \sum_{r=2}^{[p]} |v|^r |u|^{p-r} + b|v|^p,

where

a

and

b

are two constants depending only on

p

, and

[p]

is the integral part of

p

(FRV, III, §2, Exer. 6).

b) Let $(\tilde{f}_n)$ be a sequence that converges to 0 in $L^p$ for the weakened topology ( $1 < p < +\infty$ ). Show that there exists a subsequence $(f_{n_k})$ of $(f_n)$ such that, on setting $s_m = \sum_{k=1}^m f_{n_k}$ , one has

\left| \int s_{m-1}|s_{m-1}|^{p-2} f_{n_m} \, d\mu \right| \leq 1

(define the sequence

(n_k)

recursively). Deduce from this, using a) and Hölder’s inequality, that if

p > 2

there exist two constants

a, b

such that

N_1(|s_n|^p) \leq N_1(|s_{n-1}|^p) + a + bN_1(|s_{n-1}|^{p-2}),

and that if

1 < p \leq 2

there exists a constant

c

such that

N_1(|s_n|^p) \leq N_1(|s_{n-1}|^p) + c.

Conclude that

N_p(s_n) = O(n^{1/2}) \quad \text{for } p > 2 \\ N_p(s_n) = O(n^{1/p}) \quad \text{for } 1 < p \leq 2.

c) Show that the preceding results cannot in general be improved upon (cf. Exers. 19 and 20 b)).

Solution

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