37.4. MORE EXTENSION THEOREMS 1297

Lemma 37.4.1 Let H− be a half space as in 37.4.35. Let H+ be the half space in whichyn < 0 is replaced with yn > 0. Also let (y′,yn) = y

u(y′,yn

)≡{

u+ (y′,yn) if y ∈ H+

u− (y′,yn) if y ∈ H− ,

suppose u+ ∈C∞

(H+)

and u− ∈C∞

(H−), and that for l ≤ k−1,

Dlenu+(y′,0

)= Dlenu−

(y′,0

).

Then u ∈W k,p (Rn). Furthermore,

Dα u(y′,yn

)≡{

Dα u+ (y′,yn) if y ∈ H+

Dα u− (y′,yn) if y ∈ H−

Proof: Consider the following for φ ∈C∞c (Rn) and |α| ≤ k.

(−1)|α|(∫

Rn−1

∫∞

0u+Dα

φdyndy′+∫Rn−1

∫ 0

−∞

u−Dαφdyndy′

).

Integrating by parts, this yields

(−1)|α| (−1)|β |(∫

Rn−1

∫∞

0Dβ u+Dαnenφdyndy′

+∫Rn−1

∫ 0

−∞

Dβ u−Dαnenφdyndy′)

where β ≡ (α1,α2, · · ·αn−1,0) . Do integration by parts on the inside integral and by as-sumption, the boundary terms will cancel and the whole thing reduces to

(−1)|α| (−1)|β | (−1)αn

(∫Rn−1

∫∞

0Dα u+φdyndy′

+∫Rn−1

∫ 0

−∞

Dα u−φdyndy′)

=

(∫Rn−1

∫∞

0Dα u+φdyndy′+

∫Rn−1

∫ 0

−∞

Dα u−φdyndy′)

which proves the lemma.

Lemma 37.4.2 Let H− be the half space in 37.4.35 and let u ∈ C∞

(H−). Then there

exists a mapping,E : C∞

(H−)→W k,p (Rn)

and a constant, C which is independent of u ∈C∞

(H−)

such that E is linear and for alll ≤ k,

||Eu||l,p,Rn ≤C ||u||l,p,H− . (37.4.36)