16.3. RADEMACHER’S THEOREM 441

=q

q− p

(∫U0

|∇u(x+v)|q dv)1/q

=q

q− p

(∫U|∇u(z)|q dz

)1/q

Since q > p . Thus

1mp (U)

∫U|u(x)−u(z)|dz≤C

(∫U|∇u(z)|q dz

)1/q

≤C(∫

B(x,2|x−y|)|∇u(z)|q dz

)1/q

and similarly

1mp (V )

∫V|u(x)−u(z)|dz≤C

(∫B(x,2|x−y|)

|∇u(z)|q dz)1/q

From 16.4, |u(x)−u(y)| ≤C(∫

B(x,2|x−y|) |∇u(z)|q dz)1/q|x−y|1−

pq ■

Corollary 16.3.3 Let u be Lipschitz on Rp with constant K. Then there is a constant Cdepending only on p,q such that

|u(x)−u(y)| ≤C(∫

B(x,2|x−y|)|∇u(z) |qdz

)1/q(| x− y|(1−p/q)

). (16.5)

Here q > p.

Proof: Let un = u ∗ φ n where {φ n} is a mollifier as in Lemma 16.3.1. Then fromLemma 16.3.2, 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)

).

Now |∇un| = |∇u∗φ n| by Lemma 16.3.1 and this last is bounded. Also, by this lemma,∇un (z)→ ∇u(z) a.e. and un (x)→ u(x) for all x. Therefore, by the dominated conver-gence theorem, pass to the limit as n→ ∞ and obtain 16.5. ■

Note you can write 16.5 in the form

|u(x)−u(y)| ≤ C(

1|x−y|p

∫B(x,2|x−y|)

|∇u(z) |qdz)1/q

|x−y|

= Ĉ(

1mp (B(x,2 |x−y|))

∫B(x,2|x−y|)

|∇u(z) |qdz)1/q

|x−y|

Before leaving this remarkable formula, note that if you are in any situation where theabove formula holds and ∇u exists in some sense and is in Lq,q > p, then u would need tobe continuous. This is the basis for the Sobolev embedding theorem.

Here is Rademacher’s theorem.

Theorem 16.3.4 Suppose u is Lipschitz with constant K then if x is a point where∇u(x) exists,

|u(y)−u(x)−∇u(x) · (y−x)|

≤C(

1m(B(x,2 |x−y|))

∫B(x,2|x−y|)

|∇u(z)−∇u(x) |qdz)1/q

| x− y|. (16.6)

Also u is differentiable at a.e. x and also

u(x+tv)−u(x) =∫ t

0Dvu(x+ sv)ds (16.7)