38.5. THE TRACE ON THE BOUNDARY OF A HALF SPACE 1315

Corollary 38.4.3 Let u ∈ Ht (Rn) where t > m+ n2 . Then u is a.e. equal to a function of

Cmb (Rn) still denoted by u. Furthermore, there exists a constant, C independent of u such

that||u||Cm

b (Rn) ≤C ||u||Ht (Rn) .

Proof: This follows from the above lemma. Let {uk} be a sequence of functions ofS which converges to u in Ht and a.e. Then by the inequality of the above lemma, thissequence is also Cauchy in Cm

b (Rn) and taking the limit,

||u||Cmb (Rn) = lim

k→∞

||uk||Cmb (Rn) ≤C lim

k→∞

||uk||Ht (Rn) =C ||u||Ht (Rn) .

What about open sets, U?

Corollary 38.4.4 Let t > m+ n2 and let U be an open set with u ∈ Ht (U) . Then u is a.e.

equal to a function of Cm(U)

still denoted by u. Furthermore, there exists a constant, Cindependent of u such that

||u||Cm(U) ≤C ||u||Ht (U) .

Proof: Let u ∈ Ht (U) and let v ∈ Ht (Rn) such that v|U = u. Then

||u||Cm(U) ≤ ||v||Cmb (Rn) ≤C ||v||Ht (Rn) .

Now taking the inf for all such v yields

||u||Cm(U) ≤C ||u||Ht (U) .

38.5 The Trace On The Boundary Of A Half SpaceIt is important to consider the restriction of functions in a Sobolev space onto a smallerdimensional set such as the boundary of an open set.

Definition 38.5.1 For u ∈ S, define γu a function defined on Rn−1 by γu(x′) ≡ u(x′,0)where x′ ∈ Rn−1 is defined by x = (x′,xn).

The following elementary lemma featuring trig. substitutions is the basis for the proofof some of the arguments which follow.

Lemma 38.5.2 Consider the integral,∫R

(a2 + x2)−t

dx.

for a> 0 and t > 1/2. Then this integral is no more than Cta−2t+1 where Ct is some constantwhich depends on t.

Proof: If t > 1/2 the integrand is in L1 (R). This is easily seen because it is of the form1

(a2+x2)t . Now change the variable letting x = au and the result is obtained.