Espaces Lp

Lp
ΩRd
Rd
Lp(Ω)
p[1,+∞[
Lp(Ω) = fmesurables sur Ω ,RΩ|f|p<+∞
f g f g
{x∈Ω,|f(x)6=g(x)}
f g L1(Ω)
ZΩ
f=ZΩ g
f∈ L1(Ω) RΩ|f|= 0 f
Ω

L1(Ω)
f=gL1(Ω)
L1(Ω) L1(Ω)
Ω
L1(Ω) = fΩ ; ZΩ |f(x)|dx < +∞
kfkL1(Ω) =ZΩ |f(x)|dx,
L1(Ω)
L2(Ω)
L1(Ω) L2(Ω)
L2(Ω) L2(Ω)
Ω
L2(Ω) = fΩ ; ZΩ |f(x)|dx < +∞
f∈L2(Ω), g ∈L2(Ω) 7→ (f, g)L2=ZΩ f(x)g(x)dx

L2(Ω)
kfkL2(Ω) =p(f, f)L2=sZΩ |f(x)|2dx
L2(Ω)
∀f∈L2(Ω),∀g∈L2(Ω),ZΩ f(x)g(x)dx ≤ kfkL2(Ω)kgkL2(Ω)
∀f∈L2(Ω),∀g∈L2(Ω),kfgkL1(Ω) ≤ kfkL2(Ω)kgkL2(Ω)
Lp(Ω) Lp
loc(Ω)
p p ≥1
Lp(Ω)
Lp(Ω) Lp(Ω)
Ω
Lp(Ω) = 
f, Ω
ZΩ
|f(x)|pdx < +∞
.
kfkLp(Ω) =ZΩ |f(x)|pdx1/p ,
Lp(Ω)

p∈R1< p < +∞q∈
]1,∞[ 1/p + 1/q = 1 f∈Lp(Ω) g∈Lq(Ω) fg ∈L1(Ω)
kfgkL1(Ω) ≤ kfkLp(Ω) kgkLq(Ω)
p=q= 2
Ωq > p ≥1Lq(Ω) ⊂Lp(Ω)
Lp
loc(Ω)
ΩRdf
ΩKΩf
K
L1
loc(Ω) = f;f∈L1(K)K⊂Ω
Lp
loc(Ω) = {f;f∈Lp(K)K⊂Ω}.
Lp(Ω) ⊂Lp
loc(Ω)
q > p ≥1Lq
loc(Ω) ⊂Lp loc(Ω)
p≥1Lp(Rd)⊂Lp
loc(Rd)⊂L1
loc(Rd)
Lp
loc(I)fnLp loc(Ω)
f∈Lp
loc(Ω) fnf Lp loc(Ω)
K⊂Ωfn|Kf|KLp(K)
L∞(Ω)
fΩ
M
µ({x∈Ω; |f(x)| ≥ M}) = 0.

|f(x)|< M
L∞(Ω)
Ω
L∞(Ω)
L∞(Ω) L∞(Ω)
L∞(Ω) = {fΩ ; f}
kfkL∞(Ω) = inf{M≥0; µ({x∈Ω; |f(x)| ≥ M}) = 0},
L∞(Ω)
f∈L∞(Ω)
x∈Ω,|f(x)|≤kfkL∞(Ω)
kfkL∞(Ω)
fΩ
x∈Ω,|f(x)|≤kfkL∞(Ω)
f∈L1(Ω) g∈L∞(Ω) fg
L1(Ω)
kfgkL1(Ω) ≤ kfkL1(Ω)kgkL∞(Ω)
p= 1 q= +∞