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

500 J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516
For every finite collection (xi)1ik in R n , the current ❏x0,...,xk❑ ∈ Dk(R n ) is defined by
❏x0,...,xk❑,ϕ
=
Sk
ϕ
k
x0 + λixi
i=1
,x1 − x0 ∧···∧xk − x0 dλ.
Definition 2.13. A current T ∈ Dk(R n ) is a real polyhedral chain if there is (x i j )1im,1jk in
R n and (μi)1im in R such that
T =
m
i=1
i
μi x0 ,...,x i ②
k .
The set of k-dimensional real polyhedral chains is denoted by Pk(R n ). Every real polyhedral
chain has a compact support and a finite mass. Hence, 〈T,u〉 is well defined when u : R n → Λ k R n
is continuous. Moreover, if u : R n → R is continuous then 〈T,u〉∈(Λ k R n ) ′ ∼ = Λ k R n is naturally
defined.
Definition 2.14. If u : R n → R is continuous, let
u ˜Vk(R n )
= sup
P ∈Pn−k(R n )
∂P=0
M(P )1
〈P,u〉.
The seminorm u ˜Vk(R n ) measures the oscillation of the function u through its integral on
k-dimensional real polyhedral chains without boundary.
Theorem 2.15. For every n 1 and 1 k n − 1, there exists c>0 such that for every u ∈
C(R n ),
Proof. Given P in Pn−k(R n ),let
u ˜Vk(R n ) uDk(R n ) cu ˜Vk(R n ) .
ϕε(x) = P,ρε(·−x) ,
where ρε = ρ(·/ε)/ε n with ρ ∈ D(R n ), ρ 0 and
R n ρdx= 1 and where ∗ denotes the Hodge
duality between Λ k R n and Λ n−k R n . One checks that dϕε = 0, ϕε L 1 (R n ) M(P ) and
R n
Since ρε ∗ u → u uniformly as ε → 0,
uϕε dx =∗〈P,ρε ∗ u〉.
uVk(R n ) uDk(R n ).