46.2. THE TRACE ON THE BOUNDARY OF AN OPEN SET 1517

≤Cl

∑i=1||(ψ iu)||m,p,Ω ≤C

l

∑i=1||u||m,p,Ω ≤Cl ||u||m,p,Ω .

Now use the density of C∞(Ω)

in W m,p (Ω) to see that γ extends to a continuous linear mapdefined on W m,p (Ω) still called γ such that for all u ∈W m,p (Ω) ,

||γu||m− 1p ,p,∂Ω

≤Cl ||u||m,p,Ω . (46.2.3)

Also, it can be shown that γ maps W m,p (Ω) onto W m− 1p (∂Ω) . Let g ∈W m− 1

p (∂Ω).By definition, this means

h∗i (ψ ig) ∈W m− 1p (Ui) , each i

and so, using a cutoff function, there exists wi ∈W m,p (Ui× Ii) such that

γwi = h∗i (ψ ig) = h∗i (γψ ig)

Thus(H−1

i

)∗wi ∈W mp (Ω∩Wi) . Let

w≡l

∑i=1

ψ i(H−1

i)∗

wi ∈W mp (Ω)

then

γw = ∑i

γψ jγ(H−1

i)∗

wi = ∑i

γψ j(H−1

i)∗

γwi

= ∑i

γψ j(H−1

i)∗h∗i (γψ ig) = g

In addition to this, in the case where m = 1, Lemma 45.2.1 implies there exists a linearmap, R, from W 1− 1

p ,p (∂Ω) to W 1,p (Ω) which has the property that γRg = g for every

g ∈W 1− 1p ,p (∂Ω) . I show this now. Letting g ∈W 1− 1

p ,p (∂Ω) ,

g =l

∑i=1

ψ ig.

Then also,h∗i (ψ ig) ∈W 1− 1

p ,p(Rn−1)

if extended to equal 0 off Ui. From Lemma 45.2.1, there exists an extension of this toW 1,p

(Rn+

), Rh∗i (ψ ig) . Without loss of generality, assume that

Rh∗i (ψ ig) ∈W 1,p (Ui× (ai,0)) .

If not so, multiply by a suitable cut off function in the definition of R . Then the extensionis

Rg =l

∑i=1

(H−1

i)∗

Rh∗i (ψ ig) .