Regularity near the characteristic boundary for sub-laplacian operators

380 DIMITER VASSILEV
that is,
(4-3)
∫
for every ϕ ∈ C ∞ 0 ().
∫
|Xu| p−2 〈Xu, Xϕ〉 d H =
V |u| p−2 u ϕ d H,
(1) If V ∈ L Q/p (), then u ∈ L q () for every p ∗ ≤ q < ∞.
(2) If V ∈ L t () ∩ L Q/p () for some t > Q/p, then u ∈ L ∞ ().
Proof. The assumption that V ∈ L Q/p (), together with the Folland–Stein inequality,
shows that (4-3) holds true for any ϕ ∈ ˚1,p (). This can be seen by
approximating in the space ˚1,p () by a sequence of test functions ϕ n ∈ C ∞ 0 (),
which will allow us to put the limit function in the left-hand side of (4-3). On
the other hand, the Folland–Stein inequality implies that ϕ n → ϕ in L p∗ (). Set
t 0 = Q/p, and its Hölder conjugate t ′ 0 = t 0/(t 0 − 1). An easy computation gives
1
+ p − 1
t 0 p ∗ = 1 − 1 p ∗ = 1
(p ∗ ) ′ .
Hölder’s inequality then shows that V |u| p−2 u ∈ L (p∗ ) ′ (), which allows us to pass
to the limit in the right-hand side of (4-3). We turn to the proofs of (1) and (2).
(1) It is enough to prove that, if u ∈ ˚1,p () ∩ L q () with q ≥ p ∗ , then u ∈ L κq
with κ = p ∗ /p > 1. Let G(t) be a continuous, piecewise-smooth, and globally
Lipschitz function on the real line, and set
(4-4) F(u) =
∫ u
0
|G ′ (t)| p dt.
Clearly, except at finitely many points, F is a differentiable function with a bounded
and continuous derivative. From the chain rule (see [Garofalo and Nhieu 1996], for
example) there follows that F(u) ∈ ˚1,p () is a legitimate test function in (4-3).
The left-hand side, taking into account that F ′ (u) = |G ′ (u)| p , can be rewritten as
∫
∫
|Xu| p−2 〈Xu, X F(u)〉 d H = |XG(u)| p .
The Folland–Stein inequality (1-1) gives
∫
(∫
(4-5) |Xu| p−2 〈Xu, X F(u)〉 d H ≥ S p
We choose G(t) as
|G(u)| p∗ ) p/p ∗.
⎧
⎨(sign t)|t| q/p if 0 ≤ |t| ≤ l,
G(t) =
⎩
l (q/p) −1 t if l < |t|.