N. Jacquet_549

FF
α∈Rx∈R∗
+
xαeαln(x)α
R∗
+x7→ xα
R∗
+∀x∈R∗ +,d
dx xα=αxα−1.
α, β ∈Rx, y ∈R?
+
1/xα=x−α
xαyα= (xy)α
xαxβ=xα+β
(xα)β=xαβ

ln(xα) = αln x
α
lim
x→+∞xα= +∞.
0α= 0
eαln(x)
x y
xy ≤1
2(x2+y2).
p q 1
p+1 q= 1
p > 1
(x, y)∈(R+)2xy ≤1
pxp+1 qyq.
p q 1
p+1 q= 1
x1, ... , xn, y1, ... , ynn
t1, ... , tn|t1+... +tn| ≤
|t1|+... +|tn|
|x1y1+... +xnyn| ≤ |x1|p+... +|xn|p
p+|y1|q+... +|yn|q q.
∀λ∈R∗
+,|x1y1+... +xnyn| ≤ λp|x1|p+... +|xn|p
p+1 λq
|y1|q+... +|yn|q
q.
λ
|x1y1+. . . +xnyn| ≤ (|x1|p+... +|xn|p)1
p(|y1|q+...|yn|q)1 q.
x1, ..., xn, y1, ... , ynnN∗p]1,+∞[
k[[1, n]]
|xk+yk|p≤ |xk||xk+yk|p−1+|yk||xk+yk|p−1.
BY:
$
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n
X
k=1
|xk||xk+yk|p−1≤ n
X
k=1
|xk|p!1
p n
X
k=1
|xk+yk|p!1−1
p
n
X
k=1
|yk||xk+yk|p−1≤ n
X
k=1
|yk|p+!1
p n
X
k=1
|xk+yk|p!1−1
p
.
(|x1+y1|p+... +|xn+yn|p)1
p≤(|x1|p+... +|xn|p)1 p+ (|y1|p+... +|yn|p)1 p.
p∈]1,+∞[n∈N∗x1, x2, ..., xnR
k(x1, ..., xn)kp= (|x1|p+... +|xn|p)1
p
k(x1, ..., xn)k∞|x1|,|x2|, ..., |xn|
∀p∈]1,+∞[,k(x1, ..., xn)k∞≤ k(x1, ..., xn)kp≤n1
pk(x1, ..., xn)k∞.
lim
p→+∞k(x1, ..., xn)kp.
(p, q)∈(R∗
+)21/p + 1/q = 1 (a, b)∈R2a<b
f g [a;b]
∀λ∈R∗
+,Zb
a
|fg| ≤ λp
pZb
a
|f|p+λ−q
qZb
a
|g|q.
Zb
a
|f|p= 0 Zb
a
|g|q= 0 Zb
a
|fg|= 0
λ
Zb
a
|fg| ≤ Zb
a
|f|p
1
pZb
a
|g|q
1
q
.
Zb
a
|f||f+g|p−1≤Zb
a
|f|p
1
pZb
a
|f+g|p1−1
p
BY:
$
\
C

Zb
a
|g||f+g|p−1≤Zb
a
|g|p
1
pZb
a
|f+g|p1−1
p
.
Zb
a
|f+g|p
1
p
≤Zb
a
|f|p
1
p
+Zb
a
|g|p
1
p
.
BY:
$
\
C

(x−y)2≥0
y x 7→ 1
pxp+1 qyq−xy R+
xiλxiyi1
λyi
λp(|x1|p+... +|xn|p) = 1
λq(|y1|q+... +|yn|q)
|xk+yk|p=|xk+yk|p−1|xk+yk|.
q=p
p−1
|x1|, ... , |xn| |x1+y1|p−1, ... , |xn+yn|p−1
|xi| k(x1, ..., xn)k∞
(λf, 1
λg)
λ+∞0+
λ λpRb
a|f|p=λ−qRb
a|g|q
BY:
$
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C