1267

Also, (D̃α u∗φ l

)(ψ) ≡

∫U0

(∫D̃α u(y)φ l (x−y)dy

)ψ (x)dx

=∫

U0

(∫U

Dα u(y)φ l (x−y)dy)

ψ (x)dx

=∫

U0

(∫U

u(y)(Dαφ l)(x−y)dy

)ψ (x)dx

=∫

Uu(y)

∫U0

(Dαφ l)(x−y)ψ (x)dxdy

= (−1)|α|∫

Uu(y)

∫U0

φ l (x−y)(Dαψ)(x)dxdy.

It follows that Dα (ũ∗φ l)=(

D̃α u∗φ l

)as weak derivatives defined on C∞

c (U0) . Therefore,

||Dα (ũ∗φ l)−Dα u||Lp(U0)=

∣∣∣∣∣∣D̃α u∗φ l−Dα u∣∣∣∣∣∣

Lp(U0)

≤∣∣∣∣∣∣D̃α u∗φ l− D̃α u

∣∣∣∣∣∣Lp(Rn)

→ 0.

This proves the theorem.As part of the proof of the theorem, the following corollary was established.

Corollary 37.0.6 Let U0 and U be as in the above theorem. Then for all l large enoughand φ l a mollifier,

Dα (ũ∗φ l) =(

D̃α u∗φ l

)(37.0.2)

as distributions on C∞c (U0) .

Definition 37.0.7 Let U be an open set. C∞ (U) denotes the set of functions which aredefined and infinitely differentiable on U.

Note that f (x)= 1x is a function in C∞ (0,1) . However, it is not equal to the restriction to

(0,1) of some function which is in C∞c (R) . This illustrates the distinction between C∞ (U)

and C∞(U). The set, C∞

(U)

is a subset of C∞ (U) . The following theorem is known asthe Meyer Serrin theorem.

Theorem 37.0.8 (Meyer Serrin) Let U be an open subset of Rn. Then if δ > 0 and u ∈Xm,p (U) , there exists J ∈C∞ (U) such that ||J−u||m,p,U < δ .

Proof: Let · · ·Uk ⊆ Uk ⊆ Uk+1 · · · be a sequence of open subsets of U whose unionequals U such that Uk is compact for all k. Also let U−3 = U−2 = U−1 = U0 = /0. Nowdefine Vk ≡ Uk+1 \Uk−1. Thus {Vk}∞

k=1 is an open cover of U. Note the open cover islocally finite and therefore, there exists a partition of unity subordinate to this open cover,