37.2. AN EXTENSION THEOREM 1293

where the constant depends on Lip(gi) but is independent of u ∈C∞(U). Now define E as

follows.

Eu≡p

∑i=0

E i (ψ iu) .

Thus for u ∈C∞(U), it follows Eu(x) = u(x) for all x ∈U. Also,

||Eu||1,p,Rn ≤p

∑i=0

∣∣∣∣E i (ψ iu)∣∣∣∣≤ p

∑i=0

Ci ||ψ iu||1,p,Qi∩U

=p

∑i=0

Ci ||ψ iu||1,p,U ≤p

∑i=0

Ci ||u||1,p,U

≤ (p+1)p

∑i=0

Ci ||u||1,p,U ≡C ||u||1,p,U . (37.2.33)

where C depends on the ψ i and the gi but is independent of u∈C∞(U). Therefore, by den-

sity of C∞(U)

in W 1,p (U) , E has a unique continuous extension to W 1,p (U) still denotedby E satisfying the inequality determined by the ends of 37.2.33. It remains to verify thatEu(x) = u(x) a.e. for x ∈U .

Let uk → u in W 1,p (U) where uk ∈ C∞(U). Therefore, by 37.2.33, Euk → Eu in

W 1,p (Rn) . Since Euk (x) = uk (x) for each k,

||u−Eu||Lp(U) = limk→∞

||uk−Euk||Lp(U)

= limk→∞

||Euk−Euk||Lp(U) = 0

which shows u(x) = Eu(x) for a.e. x ∈U as claimed. This proves the theorem.

Definition 37.2.8 Let U be an open set. Then W m,p0 (U) is the closure of the set, C∞

c (U) inW m,p (U) .

Corollary 37.2.9 Let U be a bounded open set which has Lipschitz boundary and let W bean open set containing U. Then for each p≥ 1, there exists EW ∈L

(W 1,p (U) ,W 1,p

0 (W ))

such that EW u(x) = u(x) a.e. x ∈U.

Proof: Let ψ ∈C∞c (W ) and ψ = 1 on U. Then let EW u≡ψEu where E is the extension

operator of Theorem 37.2.7.Extension operators of the above sort exist for many open sets, U, not just for bounded

ones. In particular, the above discussion would apply to an open set, U, not necessarilybounded, if you relax the condition that the Qi must be bounded but require the existenceof a finite partition of unity {ψ i}

pi=1 having the property that ψ i and ψ i, j are uniformly

bounded for all i, j. The proof would be identical to the above. My main interest is inbounded open sets so the above theorem will suffice. Such an extension operator will bereferred to as a (1, p) extension operator.