1502 CHAPTER 45. TRACES OF SOBOLEV SPACES

Applying this to each term of the sum in 45.3.15 and adjusting the constant, it follows∫Rn

∫Rn

|u(y)−u(x)|p

|x−y|(n+pθ)dxdy≥

C (n,θ , p)n

∑i=1

∫∞

0t p(1−θ)

||u(·+ tei)−u(·)||pLp(Rn)

t pdtt

Therefore,

||u||W θ ,p(Rn)

≥ C (n,θ , p)

(||u||pLp(Rn)

+∫Rn

∫Rn

|u(y)−u(x)|p

|x−y|(n+pθ)dxdy

)1/p

This has proved most of the following theorem about the intrinsic norm.

Theorem 45.3.10 An equivalent norm for W θ ,p (Rn) is

||u||=(||u||pLp(Rn)

+∫Rn

∫Rn

|u(y)−u(x)|p

|x−y|(n+pθ)dxdy

)1/p

.

Also for any open subset of Rn

||u||=(||u||pLp(Ω)

+∫

Ω

∫Ω

|u(y)−u(x)|p

|x−y|(n+pθ)dxdy

)1/p

. (45.3.16)

is a norm.

Proof: It only remains to verify this is a norm. Recall the lp norm on R2 given by

|(x,y)|lp≡ (|x|p + |y|p)1/p

For u,v ∈W θ ,p denote by ρ (u) the expression(∫Ω

∫Ω

|u(y)−u(x)|p

|x−y|(n+pθ)dxdy

)1/p

a similar definition holding for v. Then it follows from the usual Minkowski inequality thatρ (u+ v)≤ ρ (u)+ρ (v). Then from 45.3.16

||u+ v||=(||u+ v||pLp +ρ (u+ v)p)1/p

≤ ((||u||Lp + ||v||Lp)p +(ρ (u)+ρ (v))p)

1/p