26.6. RADEMACHER’S THEOREM 959

26.6.2 Rademacher’s TheoremLemma 26.6.3 Let u be a Lipschitz continuous function which vanishes outside some com-pact set. Then there exists a unique u,i ∈ L∞ (Rn) such that

limh→0

u(·+h)−u(·)h

= u,i weak ∗ in L∞ (Rn) .

Proof: By the Lipschitz condition, the above difference quotient is bounded in L∞ byK the Lipschitz constant of u. It follows from the Banach Aloglu theorem and Corollary17.5.6 on Page 463 that there exists a subsequence hk→ 0 and g ∈ L∞ (Rn) such that

u(·+hk)−u(·)hk

→ g weak ∗ in L∞ (Rn)

Letting φ ∈C∞c (Rn) , it follows∫

gφdx = limk→∞

∫ u(·+hk)−u(·)hk

φdx =−∫

uφ ,idx

This also shows that g must vanish outside some compact set because the integral on theright shows that if sptφ does not intersect sptu, then

∫gφdx = 0. Thus g ∈ L2 (Rn). If g1

is a weak ∗ limit of another subsequence h j → 0, the same result follows. Thus for anyφ ∈C∞

c (Rn) ∫(g−g1)φdx = 0

and since C∞c (Rn) is dense in L2 (Rn) , this requires g = g1 in L2 and so they are equal a.e.

Since every sequence of h→ 0 has a subsequence which when applied to the differencequotient, always converges to the same thing, it follows the claimed limit exists. This iscalled u,i. This proves the lemma.

Lemma 26.6.4 Let u be a Lipschitz continuous function which vanishes outside a compactset and let u,i be described above. For φ ε a mollifier and uε ≡ u∗φ ε ,

uε,i = u,i ∗φ ε

where the symbol uε,i means the usual partial derivative with respect to the ith variable.Also for any p > n,

uε,i→ u,i in Lp (Rn) .

Proof: This follows from a computation and Lemma 26.6.3.

uε,i (x)≡ limh→0

∫ u(x−y+hei)−u(x−y)h

φ ε (y)dy

= limh→0

∫ u(z+hei)−u(z)h

φ ε (x− z)dz

=∫

u,i (z)φ ε (x− z)dz = u,i ∗φ ε (x)