Regularity near the characteristic boundary for sub-laplacian operators

376 DIMITER VASSILEV
Continuing with the proof that the dichotomy case does not occur, we use the
definition of I , see (2-1) and (3-2), together with the above inequalities, to get
‖v n ‖ p˚1,p (G) = ‖ϕ mv n ‖ p˚1,p (G) + ‖(1 − ϕ n)v n ‖ p˚1,p (G) + ε n
≥ I ( ‖ϕ m v n ‖ p L p∗ (G) + ‖(1 − ϕ n)v n ‖ p L (G))
+ εn
p∗
((∫ ) p/p ∗ (∫
≥ I dν n +
B Rn (g n )
((∫
≥ I
Letting n → ∞, we obtain
dνn
1
G
) p/p ∗
(∫
+ dνn
2
G
) p/p ∗)
dν n + ε n
G\B Rn (g n )
) p/p ∗)
+ ε n .
I = lim
n→∞ ‖v n‖ p˚1,p (G) ≥ I ( λ p/p∗ + (1−λ) p/p∗ )
− εI,
which is a contradiction with the choice of ε in (3-27), and hence the dichotomy
case of Lemma 3.3 cannot occur. The proof of the theorem is finished. □
3B. The best constant in the presence of symmetries. We consider here the same
problem as before, but we restrict the class of test functions.
Definition 3.6. Let G be a Carnot group with Lie algebra g = V 1 ⊕ V 2 · · · ⊕ V n .
We say that a function U : G → has partial symmetry with respect to g 0 if there
exist an element g 0 ∈ G such that for every g = exp(ξ 1 +ξ 2 +· · ·+ξ n ) ∈ G one has
U(g 0 g) = u ( |ξ 1 (g)|, . . . , |ξ n−1 (g)|, ξ n (g) ) ,
for some function u : [0, ∞) × · · · × [0, ∞) × V n → .
A function U is said to have cylindrical symmetry if there exist g 0 ∈ G and
ϕ : [0, ∞) × · · · × [0, ∞) → for which
for every g ∈ G.
U(g 0 g) = ϕ ( |ξ 1 (g)|, |ξ 2 (g)|, . . . , |ξ n (g)| ) ,
We also define the spaces ˚1,p ps (G) and ˚1,p (G) by
(3-36)
(3-37)
cyl
˚ 1,p def
ps (G) = { u ∈ ˚1,p () ∣ u(g) = u ( |ξ 1 (g)|, . . . , |ξ n−1 (g)|, ξ n (g) )} ,
˚ 1,p def
cyl
(G) = { u ∈ ˚1,p () ∣ u(g) = u ( |ξ 1 (g)|, |ξ 2 (g)|, . . . , |ξ n (g)| )} .
The effect of the symmetries (see also [Lions 1985b]) is manifested in the fact
that, if the limit measure given by Lemma 3.5 concentrates at a point, then it
must concentrate on the whole orbit of the group of symmetries. Therefore, in
the cylindrical case there could be no points of concentration except at the origin,