Regularity near the characteristic boundary for sub-laplacian operators

382 DIMITER VASSILEV
(2) It is enough to prove that the L q () norms of u are uniformly bounded by
some sufficiently large but fixed L q 0
norm of u, q 0 ≥ p ∗ , which is finite from (1).
We shall do this by iteration. We use the function F(u) from part (1) in the weak
form (4-3) of our equation. The left-hand side is estimated from below as before,
in (4-5). This time, though, we use Hölder’s inequality to estimate from above the
right-hand side:
∫
V |u| p−2 ∥
(4-8)
uF(u) d H ≤ ‖V ‖ t |u| p−1 F(u) ∥ t ′
With the estimate from below, we come to
Letting l → ∞, we obtain
(4-9)
≤ ‖V ‖ t
∥ ∥ C(q)|G(u)| p∥ ∥
t ′ ≤ C(q)‖V ‖ t ‖u‖ q qt ′ .
S p ‖G(u)‖ p p ∗ ≤ C(q)‖V ‖ t ‖u‖ q qt ′ .
∥ |u|
q/p ∥ ∥ p p ∗ ≤ C(q)
S p
‖V ‖ t ‖u‖ q qt ′ .
Set δ = p ∗ /(pt ′ ). The assumption t > Q/p implies δ > 1, since the latter is equivalent
to t ′ < p ∗ /p = t ′ 0 , as t 0 = Q/p. With this notation we can rewrite (4-9) as
( C(q)
) 1/q‖V
1/q
(4-10) ‖u‖ δqt ′ ≤
‖ t ‖u‖ qt ′.
S p
Recall that C(q) ≤ Cq p−1 . At this point, we define q 0 = p ∗ t ′ and q k = δ k q 0 . After
a simple induction, we obtain
k−1
∏ ( p−1) t
(4-11) ‖u‖ qk ≤ Cq ′ /q ∑ j
j
‖V ‖ t′ k−1
j=0 1/q j
t ‖u‖ q0 .
j=0
We observe that the right-hand side is finite,
(4-12)
∞∑
j=0
1
= 1 ∑
∞ 1
q j
q 0 δ j < ∞
j=1
and
∞∑
j=1
log q j
q j
< ∞,
because δ > 1. Letting j → ∞, we obtain
‖u‖ ∞ ≤ C‖u‖ q0 .
□
Remark 4.2. When is a bounded open set, we trivially have V ∈ L Q/p ()
whenever V ∈ L t () with t > Q/p. Also, in this case one can obtain a uniform
estimate of the L ∞ () norm of u by its L p∗ () norm that does not depend on the
distribution function of V , as we had in the preceding theorem. This can even be