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

J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 501
The converse inequality is based on the deformation Theorem for currents [10,17]. It states
that given T ∈ Dk(R n ) with M(T ) + M(∂T ) < ∞, for every ε>0, there exists P ∈ Pk(R n ),
S ∈ Dk(R n ) and R ∈ Dk+1(R n ) such that
with
and
T − P = ∂R + S,
M(P ) cM(T ), M(∂P ) cM(∂T ),
M(R) cεM(T ), M(S) cεM(∂T ),
supp P ∪ supp R ⊂ x ∈ R n :dist(x, supp T)0 in Pk(R n ) with ∂Pε = 0 and supp Pε ⊂ U such that for every
u ∈ C(R n ; R n ),
and
M(Pε) cM(T ), (2.2)
〈Pε,u〉→〈T,u〉 (2.3)
as ε → 0.
Now given ϕ ∈ D#(Rn ; ΛkRn ), consider T ∈ Dn−k(Rn ) defined by
〈T,v〉= ϕ ∧ vdx.
R n
Since T has compact support, ∂T = 0 and M(T ) ϕ L 1 (R n ) , there is a sequence (Pε)ε>0 in
Pn−k(R n ) such that ∂Pε = 0, supp Pε ⊂ U, (2.2) and (2.3), where U is a fixed open bounded set
such that supp T ⊂ U. ✷
3. Basic properties of Dk(R n )
3.1. Mutual injections
The collection of spaces Dk(R n ) is a decreasing sequence of spaces.
Theorem 3.1. Let k ℓ.Ifu ∈ Dℓ(R n ), then u ∈ Dk(R n ), and
where C does not depend on u.
uDk(R n ) CuDℓ(R n ),