1282 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

First of all, for u ∈ K and x ∈ Rn,

|u∗φ k (x)|p ≤

(∫|u(x−y)φ k (y)|dy

)p

=

(∫|u(y)φ k (x−y)|dy

)p

≤∫|u(y)|p φ k (x−y)dy

≤(

supz∈Rn

φ k (z))∫|u(y)|dy≤M

(supz∈Rn

φ k (z))

showing the functions in Kk are uniformly bounded.Next suppose x,x1 ∈ Kk and consider

|u∗φ k (x)−u∗φ k (x1)|

≤∫|u(x−y)−u(x1−y)|φ k (y)dy

≤(∫|u(x−y)−u(x1−y)|p dy

)1/p(∫φ k (y)

q dy)q

which by assumption 37.1.17 is small independent of the choice of u whenever |x−x1| issmall enough. Note that k is fixed in the above. Therefore, the set, Kk is precompact inC(G)

thanks to the Ascoli Arzela theorem. Next consider how well u ∈ K is approximatedby u∗φ k in Lp (Rn) . By Minkowski’s inequality,(∫

|u(x)−u∗φ k (x)|p dx)1/p

≤(∫ (∫

|u(x)−u(x−y)|φ k (y)dy)p

dx)1/p

≤∫

B(0, 1k )

φ k (y)(∫|u(x)−u(x−y)|p dx

)1/p

dy.

Now let η > 0 be given. From 37.1.17 there exists k large enough that for all u ∈ K,∫B(0, 1

k )φ k (y)

(∫|u(x)−u(x−y)|p dx

)1/p

dy≤∫

B(0, 1k )

φ k (y)η dy = η .

Now let ε > 0 be given and let δ and G correspond to ε as given in the hypotheses andlet 1/k < δ and also k is large enough that for all u ∈ K,

||u−u∗φ k||p < ε

as in the above inequality. By the Ascoli Arzela theorem there exists an(ε

m(G+B(0,1)

))1/p