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

J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 499
One has fεL1 (Rn ) 1 and div fε = 0, therefore
ufε dx
uDn−1(Rn ).
R n
The proof continues as for the first part of Proposition 2.10.
The converse inequality comes from on a result of Smirnov [18], which states that for
every R>0 and for every f ∈ D(B(0,R); Rn ) there exists (γ ℓ m )1m,ℓ in C1 (S1 ; B(0,R)) and
(λℓ m )1m,ℓ in R such that for every m 1,
λ ℓ
m
˙γ ℓ
m L1 (S1 f ) L1 (Rn ) ,
and for every u ∈ C(B(0,R))
as m →∞. ✷
ℓ1
∞
ℓ=1
λ ℓ m
2.6. Geometric characterization of Vk(R n )
S 1
u γ ℓ m (t) ˙γ ℓ
m dt →
R n
uf d x
The characterization of the seminorm ·Dk(R n ) of Proposition 2.10, relies essentially on the
fact that the seminorm could be evaluated by considering differential of scalar functions, while
in Proposition 2.11 it relied on the decomposition result of Smirnov. Those facts do not hold
anymore for 1