37.1. EMBEDDING THEOREMS FOR W m,p (Rn) 1285

Lemma 37.1.16 Let U be a bounded open set and define for p > 1

S≡{

u ∈W 1,1 (U)∩Lp (U) : ||u||1,1,U + ||u||Lp(U) ≤M}

(37.1.19)

and let φ ∈C∞c (U) and

S1 ≡ {uφ : u ∈ S} . (37.1.20)

Then S1 is precompact in Lq (U) where 1≤ q < p.

Proof: This depends on Theorem 37.1.14. The second condition is satisfied by takingG≡ spt(φ). Thus, for w ∈ S1, ∫

U\G|w(x)|q dx = 0 < ε

p.

It remains to satisfy the first condition. It is necessary to verify there exists δ > 0 such thatif |v|< δ , then ∫

Rn

∣∣∣φ̃u(x+v)− φ̃u(x)∣∣∣q dx < ε

p. (37.1.21)

Let spt(φ)∪ (spt(φ)−v)≡ Gv. Now if h is any measurable function, and if θ ∈ (0,1)is chosen small enough that θq < 1,∫

Gv|h|q dx =

∫Gv|h|θq |h|(1−θ)q dx

≤(∫

Gv|h|dx

)θq(∫Gv

(|h|(1−θ)q

) 11−θq

)1−θq

=

(∫Gv|h|dx

)θq(∫Gv|h|

(1−θ)q1−θq

)1−θq

. (37.1.22)

Now let θ also be small enough that there exists r > 1 such that

r(1−θ)q1−θq

= p

and use Holder’s inequality in the last factor of the right side of 37.1.22. Then 37.1.22 isdominated by (∫

Gv|h|dx

)θq(∫Gv|h|p) 1−θq

r(∫

Gv1dx)1/r′

= C(||h||Lp(Gv)

,mn (Gv))(∫

Gv|h|dx

)θq

.

Therefore, for u ∈ S,∫Rn

∣∣∣φ̃u(x+v)− φ̃u(x)∣∣∣q dx =

∫Gv|φu(x+v)−φu(x)|q dx≤