Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

502 J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516
Proof. Let ϕ ∈ D#(Rn ; ΛkRn ).Ifα ∈ Λℓ−kRn , then α ∧ ϕ ∈ D#(Rn ; ΛℓRn ). Therefore
uα∧ ϕdx
uDℓ(Rn )α ∧ ϕL1 (Rn ) uDℓ(Rn )|α|ϕL1 (Rn ) .
R n
Taking the supremum over α ∈ Λ ℓ−k R n with |α| 1 leads to the conclusion. ✷
3.2. Extension theory
If n 0 are independent of u and U.
R n
R N−n
ϕ(x,y)dy dx.
cU Dk(R N ) uDk(R n ) CU Dk(R N ) ,
Proof. By induction, it is sufficient to consider the case N = n + 1.
First let us estimate U Dk(R N ) . Consider Φ ∈ D#(R N ; Λ k R N ). It can be written as
Φ = Φ0 + Φ1 ∧ ωN,
where Φ0 ∈ D#(R N ; Λ k R n ) and Φ1 ∈ D#(R N ; Λ k−1 R n ). Define
ϕ(x) =
For m = 1, 2,
and
R
Φ(x,t)dt, ϕ0(x) =
dϕm(x) =
R n
n
i=1
R
ωi ∧ ∂ϕm
∂xi
ϕm dx =
Φ0(x, t) dt, ϕ1(x) =
=
Rn R
R
N
i=1
ωi ∧ ∂Φm
∂xi
Φm dt dx = 0.
R
dt = 0
Φ1(x, t) dt.