Regularity near the characteristic boundary for sub-laplacian operators

370 DIMITER VASSILEV
(2) Vanishing: For all R > 0, we have
( ∫
lim
n→∞
sup
g∈G
)
dν n = 0.
B(g,R)
(3) Dichotomy: There exists a λ with 0 < λ < 1, such that, for every ε > 0, there
exist R > 0 and a sequence (g n ) with the property: Given R ′ > R, there exist
nonnegative measures νn 1 and ν2 n for which
0 ≤ ν 1 n + ν2 n ≤ ν n,
supp νn 1 ⊂ B(g n, R) and supp νn 2 ⊂ G \ B(g n, R ′ ),
∫
∫
∣
∣
∣λ − ∣ + ∣(1 − λ) − ∣ ≤ ε.
ν 1 n
Remark 3.4. Using diagonal subsequences, we can also achieve
ν 2 n
supp νn 1 ⊂ B R n
(g n ) and supp νn 2 ⊂ G \ B 2R n
(g n ),
∫
∫
∣
∣
lim ∣λ − ∣ + ∣(1 − λ) − ∣ = 0.
n→∞
νn
1
G
νn
2
G
Lemma 3.5. Suppose u n ⇁ u weak-∗ in ˚1,p (G), µ n = |Xu n | p d H ⇁ µ, and
ν n = |u n | p∗ d H ⇁ ν weak-∗ in measure, where µ and ν are bounded nonnegative
measures on G. There exist points g j ∈ G and real numbers d j ≥ 0 and e j ≥ 0, at
most countably many nonzero, such that
ν = |u| p∗ + ∑ j
d j δ gj
µ ≥ |Xu| p d H + ∑ j
e j δ gj
I d j
p/p ∗ ≤ e j ,
where I is the constant from (3-2). In particular,
∑
dj
p/p ∗ < ∞.
Proof of Theorem 3.1. Since p > 1, we can assume that u n ⇁ u weak-∗ in ˚1,p (G)
for some u ∈ ˚1,p (G), by regarding ˚1,p (G) as the dual of 1,p′ (G), where p ′ is
the Hölder conjugate of p. From the Folland–Stein embedding theorem, this is also
true for the weak-∗ convergence in L p∗ (G). We can also assume the minimizing
sequence is a.e. pointwise convergent on G. This follows easily from Rellich’s
theorem, applied successively to an exhaustion of G by an increasing sequence of
Carnot–Carathéodory balls.