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

504 J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516
Then u ∈ Vk(R n ), and
uDk(R n ) U DK(R N ) .
Proof. By induction, we can assume N = n + 1. Let ϕ ∈ D#(Rn ; ΛkRn ),letρ∈D(R) such
that ρ 0 and
R ρdt = 1 and let ρε(t) = ρ(t/ε)/ε.LetΦε(x, t) = ρε(t)ψ(x) ∧ ωN . Since
Φε ∈ D#(RN ; ΛKRN ) and u is continuous,
uϕ dx
= lim
uΦε dt dx
R n
ε→0
Rn R
4. Examples of functions in Dk(R n )
4.1. Sobolev spaces
UDK (RN ) lim
ε→0 ΦεL1 (RN ) UDK (RN ) ϕL1 (Rn ) . ✷
The first class of functions in the space Dk(R n ) are functions in critical Sobolev spaces, which
motivated the definition.
Theorem 4.1. (Bourgain and Brezis [3]) If u ∈ W s,p (R n ), p>1 and sp = n, then for every
1 k n − 1, u ∈ Dk(R n ), and
uDk(R n ) Ck,s,puW s,p (R n ).
The seminorm on the right-hand side is the Sobolev semi-norm. For 0