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

J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 493
While VMO(R3 ) ⊂ D1(R3 ), one has the following continuous embeddings:
3
D2 R 3
⊂ D1 R ⊂ BMO R 3 .
The first embedding is a consequence of (1.6), and the second of the duality between BMO(R3 )
and the Hardy space H1 (R3 ), and of a decomposition of every function in H1 (R3 ) as a sum of
some components of curl-free vector-fields.
If u ∈ D1(R2 ), its extension U(x,y) = u(x) to R3 is in D1(R3 ).ItwouldbeinD1(R3 )
if and only if U was bounded. On the other hand, if u ∈ D2(R3 ) is continuous, one has
the trace inequality u| R2D1(R 2 ) CuD2(R3 ) . The problem whether the trace inequalities
u| R2BMO(R2 ) CuD1(R3 ) and u|RBMO(R) CuD2(R3 ) hold is open.
The seminorm ·D1(R3 ) can also be characterized geometrically: by the co-area formula, for
every u ∈ C(R3 ),
uD1(R3 1
) = sup
Ω H2
(∂Ω) u(y)ν(y)dH 2
(y)
,
where the supremum is taken over bounded domains Ω ⊂ R 3 with a smooth connected boundary,
ν(y) is the unit exterior normal vector to the boundary at y ∈ ∂Ω, and H 2 is the two-dimensional
Hausdorff measure.
1.3. Integrals along differential forms
In higher dimensions, the generalization of (1.1) corresponding to (1.5) in R3 is expressed
with differential forms: if 1 k n − 1, then, for every compactly supported smooth
k-differential form ϕ ∈ D(Rn ; ΛkRn ) and for every u ∈ Ws,p (Rn ) with p 1 and sp = n, if
dϕ = 0, then
uϕ dx
Cs,pϕL1 (Rn ) uWs,p (Rn ). (1.7)
R n
The previous definitions of Dk(Rn ) are generalized as follows. For 1 k n − 1, we define
the seminorm
uϕ dx
and the vector space
∂Ω
uDk(R n ) = sup
ϕ∈D(R n ;Λ k R n )
dϕ=0
ϕ L 1 (R n ) 1
R n
n
Dk R = u ∈ D ′ R n : uDk(Rn ) < ∞ .
By (1.7), Ws,p (Rn ) ⊂ Dk(Rn ). These spaces Dk(Rn ) also contain other functions, such as
log( k+1 i=1 x2 i ).
The spaces Dk(Rn ) contain neither BMO(Rn ) nor VMO(Rn ). Our main result is that Dk(Rn )
is embedded in BMO(Rn ). We first show that Dk(Rn ) is embedded in D1(Rn ), then we prove