962 CHAPTER 26. INTEGRALS AND DERIVATIVES

Proof: If the weak limit exists, then the difference quotients must be bounded. Thisfollows from the uniform boundedness theorem, Theorem 17.1.8. Here is why. Denote thedifference quotient by Dh to save space. Weak convergence requires

∫Dh f →

∫u.i f for all

f ∈ Lp′ . Could there exist hk such that∣∣∣∣Dhk

∣∣∣∣Lp → ∞? Not unless a subsequence satisfies

hk→ 0 because if this sequence is bounded away from 0, the formula for Dh will yield thedifference quotients are bounded. However, if hk→ 0, then for each f ∈ Lp′ ,

supk

∫Dhk f < ∞

because in fact, limk→∞

∫Dhk f exists so it must be bounded. Now Dhk can be considered in(

Lp′)′

and this shows it is pointwise bounded on Lp′ . Therefore, Dhk is bounded in(

Lp′)′

but the norm on this is the same as the norm in Lp. Thus Dhk is bounded after all.Conversely, if the difference quotients are bounded, the same argument used earlier,

involving convergence of a subsequence, this time coming from the Eberlein Smulian the-orem, Theorem 17.5.12 and showing that every subsequence converges to the same thing,shows the difference quotients converge weakly in Lp (Rn) to something we can call u,i.This proves the lemma.

Definition 26.6.9 A function f ∈ Lp (Rn) is said to have weak partial derivatives in Lp (Rn)

if the difference quotients u(·+hei)−u(·)h for each i = 1,2, · · · ,n are bounded for h ̸= 0. If

f ∈ Lp (Rn;Rm), it has weak partial derivatives in Lp (Rn;Rm) if each component functionhas weak partial derivatives in Lp (Rn) .

This following theorem may also be referred to as Rademacher’s theorem.

Theorem 26.6.10 Let h be in Lp (Rn;Rm) , p > n, and suppose it has weak derivativesh,i ∈ Lp (Rn;Rm) for i = 1, · · · ,n. Then Dh(x) exists a.e. and h is almost everywhere equalto a continuous function. Also if φ ε is a mollifier,

(h∗φ ε),i = h,i ∗φ ε , (h∗φ ε),i→ h,i

in Lp (Rn;Rm) .

Proof: As before,

(h∗φ ε),i (x)≡ limh→0

∫ h(x+hei−y)−h(x−y)h

φ ε (y)dy

= limh→0

∫ h(z+hei)−h(z)h

φ ε (x− z)dy≡∫

h,i (z)φ ε (x−y)dy

= h,i ∗φ ε (x)

and now (h∗φ ε),i→ h,i follows as before from a use of Minkowski’s inequality. Letting ube one of the component functions of h, Morrey’s inequality holds for uε ≡ u∗φ ε . Thus

|uε (x)−uε (y)| ≤C(∫

B(x,2|x−y|)|∇uε (z)|p dz

)1/p(|x−y|1−n/p

)