26.5. RADEMACHER’S THEOREM 953
But this is just the polar coordinates description of what follows.
=C∫
U0
|∇u(x+ z)||z|p−1 dz
≤C(∫
U0
|∇u(x+ z)|q dz)1/q(∫
U0
|z|q′−pq′
)1/q′
= C(∫
U|∇u(z)|q dz
)1/q(∫Sp−1
∫ r
0ρ
q′−pq′ρ
p−1dρdσ
)(q−1)/q
= C(∫
U|∇u(z)|q dz
)1/q(∫
Sp−1
∫ r
0
1
ρp−1q−1
dρdσ
)(q−1)/q
= C(
q−1q− p
)(q−1)/q(∫U|∇u(z)|q dz
)1/q
r1− pq
= C(
q−1q− p
)(q−1)/q(∫U|∇u(z)|q dz
)1/q
|x−y|1−pq
Similarly,
1m(V )
∫U|u(y)−u(z)|dz≤C
(q−1q− p
)(q−1)/q(∫V|∇u(z)|q dz
)1/q
|x−y|1−pq
Therefore,
|u(x)−u(y)| ≤C(
q−1q− p
)(q−1)/q(∫B(x,2|x−y|)
|∇u(z)|q dz)1/q
|x−y|1−pq
because B(x,2 |x−y|)⊇V ∪U.
Corollary 26.5.4 Let u be Lipschitz on Rp with constant K. Then there is a constant Cdepending only on p such that
|u(x)−u(y)| ≤C(∫
B(x,2|x−y|)|∇u(z) |qdz
)1/q(| x− y|(1−p/q)
). (26.5.13)
Here q > p.
Proof: Let un = u ∗ φ n where {φ n} is a mollifier as in Lemma 26.5.2. Then fromLemma 26.5.3, there is a constant depending only on p such that
|un (x)−un (y)| ≤C(∫
B(x,2|x−y|)|∇un (z) |qdz
)1/q(| x− y|(1−p/q)
).