16.3. RADEMACHER’S THEOREM 439

This proves 16.1.

|un (x)−u(x)| ≤∫Rp|u(x−y)−u(x)|φ n (y)dmp (y)

=∫

B(0, 1n )|u(x−y)−u(x)|φ n (y)dmp (y)

by uniform continuity of u coming from the Lipschitz condition, when n is large enough,this is no larger than

∫Rp εφ n (y)dmp (y) = ε and so uniform convergence holds.

Now consider the last claim. From the first part,

|unxi (x)−uxi (x)| =

∣∣∣∣∣∫

B(0, 1n )

uxi (x−y)φ n (y)dmp (y)−uxi (x)

∣∣∣∣∣=

∣∣∣∣∣∫

B(x, 1n )

uxi (z)φ n (x−z)dmp (z)−uxi (x)

∣∣∣∣∣|unxi (x)−uxi (x)| ≤

∫Rp|uxi (x−y)−uxi (x)|φ n (y)dmp (y)

=∫

B(0, 1n )|uxi (x−y)−uxi (x)|φ n (y)dmp (y)

=∫

B(x, 1n )|uxi (z)−uxi (x)|φ n (x−z)dmp (z)

≤ np∫

B(x, 1n )|uxi (z)−uxi (x)|φ (n(x−z))dmp (z)

≤ Cmp(B(0, 1

n

)) ∫B(x, 1

n )|uxi (z)−uxi (x)|dmp (z)

which converges to 0 for a.e. x, in fact at any Lebesgue point. This is because uxi isbounded by K and so is in L1

loc. ■Note that the above holds just as well if u has values in some Rm and the same proof

would work, replacing |·| with ∥·∥ or the Euclidean norm |·|.The following lemma gives an interesting inequality due to Morrey. To simplify nota-

tion dz will mean dmp (z).

Lemma 16.3.2 Let u be a C1 function on Rp. Then there exists a constant C, dependingonly on p such that for any x, y ∈ Rp,

|u(x)−u(y)| ≤C(∫

B(x,2|x−y|)|∇u(z) |qdz

)1/q(| x− y|(1−p/q)

). (16.3)

Here q > p and C is some constant depending on p,q.

Proof: In the argument C will be a generic constant which depends on p,q. Considerthe following picture.