1484 CHAPTER 45. TRACES OF SOBOLEV SPACES

Definition 45.1.3 We define the trace,

γ : W 1,p (Rn+

)→ Lp (Rn−1)

as follows. γφ (x′)≡ φ (x′,0) whenever φ ∈C∞(Rn+

). For u ∈W 1,p

(Rn+

), we define γu≡

limk→∞ γφ k in Lp(Rn−1

)where φ k → u in W 1,p

(Rn+

). Then the above lemma shows this

is well defined.

Also from this lemma we obtain a constant, C such that

||φ ||0,p,Rn−1 ≤C ||φ ||1,p,Rn+

and the same constant holds for all u ∈W 1,p(Rn+

).

From the definition of the norm in the trace space, if f ∈ C∞(Rn+

), and letting θ =

1− 1p , it follows from the definition

||γ f ||1− 1p ,p,Rn−1

≤ max

((∫∞

0

(t1/p || f (t)||1,p,Rn−1

)p dtt

)1/p

,

(∫∞

0

(t1/p ∣∣∣∣ f ′ (t)∣∣∣∣0,p,Rn−1

)p dtt

)1/p)

≤ C || f ||1,p,Rn+.

Thus, if f ∈W 1,p(Rn+

), define γ f ∈W 1− 1

p ,p(Rn−1

)according to the rule,

γ f = limk→∞

γφ k,

where φ k → f in W 1,p(Rn+

)and φ k ∈ C∞

(Rn+

). This shows the continuity part of the

following lemma.

Lemma 45.1.4 The trace map, γ, is a continuous map from W 1,p(Rn+

)onto

W 1− 1p ,p(Rn−1) .

Furthermore, for f ∈W 1,p(Rn+

),

γ f = f (0) = limt→0+

f (t)

the limit taking place in Lp(Rn−1

).

Proof: It remains to verify γ is onto along with the displayed equation. But by defi-nition, things in W 1− 1

p ,p(Rn−1

)are limt→0+ f (t) where f ∈ Lp

(0,∞;W 1,p

(Rn−1

)), and

f ′ ∈ Lp(0,∞;Lp

(Rn−1

)), the limit taking place in

W 1,p (Rn−1)+Lp (Rn−1)= Lp (Rn−1) ,