Exercise 18, § 6

Integration, IV, p. 139

Bourbaki marks this one as harder than the rest.

Let $X$ be a locally compact space, $\mu$ a positive measure on $X$ , and $f$ a $\mu$ -measurable numerical function.

a) Let $(a_n)_{n \in \mathbf{Z}}$ be a sequence of real numbers, $\lambda$ a finite real number; for every $n \in \mathbf{Z}$ , set

u_n(x) = a_{n+\lfloor \lambda \log |f(x)| \rfloor} f(x) \quad \text{if } f(x) \neq 0 \text{ and } f(x) \neq \pm \infty \\ u_n(x) = 0 \quad \text{for the other values of } x \in X

(where

[t]

denotes the integral part of the finite real number

t

). Show that the

u_n

are

\mu

-measurable and that, for every finite real number

c

such that

c\lambda < 1

\sum_{n=-\infty}^{+\infty} \int^* |u_n(x)| e^{cx} d\mu(x) \leq e^{|c|} \int^* |f(x)|^{1-c\lambda} d\mu(x) \cdot \left( \sum_{n=-\infty}^{+\infty} e^{cn}|a_n| \right).

b) Let $p', p''$ be two finite numbers $> 0$ , $t$ a real number such that $0 < t < 1$ , and let $p$ be the real number defined by $1/p = (1-t)/p' + t/p''$ . For every number $\alpha > 0$ , set

1/K_\alpha = \inf \left( \sum_{n=-\infty}^{+\infty} |e^{\alpha tn} a_n|^{p'} \right)^{(1-t)/p'} \left( \sum_{n=-\infty}^{+\infty} |e^{-\alpha(1-t)n} a_n|^{p''} \right)^{t/p''},

the infimum being taken over all the absolutely convergent series

(a_n)_{n \in \mathbf{Z}}

of finite real numbers such that

\sum_{n=-\infty}^{+\infty} a_n = 1

. Show that ()
$N_p(f) \leq K_\alpha \cdot \inf F(u)$
(*)

N_p(f) \geq K_\alpha e^{-\alpha} \cdot \inf F(u),

where

u = (u_n)_{n \in \mathbf{Z}}

runs over the set of sequence of functions belonging to

\mathscr{L}^{p'} \cap \mathscr{L}^{p''}

such that the series

(u_n(x))_{n \in \mathbf{Z}}

is almost everywhere absolutely convergent with sum

f(x)

; for every sequence

u

having these properties, one sets

F(u) = \left( \sum_{n=-\infty}^{+\infty} (N_{p'}(e^{\alpha tn} u_n))^{p'} \right)^{(1-t)/p'} \left( \sum_{n=-\infty}^{+\infty} (N_{p''}(e^{-\alpha(1-t)n} u_n))^{p''} \right)^{t/p''}

and in the formulas () and (*) the infimum is taken over the set of sequences

u = (u_n)

having the preceding properties. (To prove (), use Hölder’s inequality; to prove (*), use a) twice with suitable choices of

c

and

\lambda

.) In particular,

K_\alpha

is finite for all

\alpha > 0

c) Let $p', p'', q', q''$ be finite numbers $\geq 1$ , $t$ a number such that $0 < t < 1$ , and let

\frac{1}{p} = \frac{1-t}{p'} + \frac{t}{p''}, \quad \frac{1}{q} = \frac{1-t}{q'} + \frac{t}{q''}.

Let

Y

be a second locally compact space,

\nu

a positive measure on

Y

, and let

w

be a linear mapping of

\mathscr{H}(X; \mathbf{R})

into the vector space (not topological)

\mathscr{S}(Y, \nu; \mathbf{R})

\nu

-measurable finite numerical functions on

Y

. Suppose that:

1^\circ

w

maps

\mathscr{H}(X; \mathbf{R})

into

\mathscr{L}^{q'}(Y; \mathbf{R}) \cap \mathscr{L}^{q''}(Y; \mathbf{R})

;

2^\circ

for every function

f \in \mathscr{H}(X; \mathbf{R})

N_{q'}(w(f)) \leq M' N_{p'}(f) \quad \text{and} \quad N_{q''}(w(f)) \leq M'' N_{p''}(f).

From this, conclude that

w

also maps

\mathscr{H}(X; \mathbf{R})

into

\mathscr{L}^q(Y; \mathbf{R})

and that

N_q(w(f)) \leq M \cdot N_p(f)

with

M \leq {M'}^{1-t} {M''}^t

(M. Riesz’s inequality). (Write

f

in the form

\sum_n u_n

as in b), and consider the series with general term

w(u_n)

; use the inequalities () and (*) of b).)

Solution

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