Kenneth Kuttler

1284 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

depending only on the indicated quantities such that whenever v ∈ Rn with

|v|< dist(spt(φ) ,UC) ,

it follows that ∫Rn

∣∣∣φ̃u(x+v)− φ̃u(x)∣∣∣dx≤C

(φ , ||u||1,1,U

)|v| .

Proof: First suppose u ∈ C∞(U). Then for any x ∈ spt(φ)∪ (spt(φ)−v) ≡ Gv, the

chain rule implies

|φu(x+v)−φu(x)| ≤∫ 1

∑i=1

∣∣∣(φu),i (x+ tv)vi

∣∣∣dt

≤∫ 1

∑i=1

∣∣(φ ,iu+u,iφ)(x+ tv)

∣∣dt |v| .

Therefore, for such u, ∫Rn

∣∣∣φ̃u(x+v)− φ̃u(x)∣∣∣dx

=∫

Gv|φu(x+v)−φu(x)|dx

≤∫

∫ 1

∑i=1

∣∣(φ ,iu+u,iφ)(x+ tv)

∣∣dtdx |v|

≤∫ 1

∫Gv

∑i=1

∣∣(φ ,iu+u,iφ)(x+ tv)

∣∣dxdt |v|

≤ C(

φ , ||u||1,1,U)|v|

where C is a continuous function of ||u||1,1,U . Now for general u ∈W 1,1 (U) , let uk→ u inW 1,1 (U) where uk ∈C∞

(U). Then for |v|< dist

(spt(φ) ,UC

),∫

∣∣∣φ̃u(x+v)− φ̃u(x)∣∣∣dx

=∫

Gv|φu(x+v)−φu(x)|dx

= limk→∞

∫Gv|φuk (x+v)−φuk (x)|dx

≤ limk→∞

φ , ||uk||1,1,U)|v|

= C(

φ , ||u||1,1,U)|v| .

This proves the lemma.

1284 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACESdepending only on the indicated quantities such that whenever v € RR" with|v| < dist (spt(@) ,U),it follows that[, \u(x-+v) — u(x) |dx<C (6, la aw) Iv].Proof: First suppose u € C” (U). Then for any x € spt() U (spt($) —v) = Gy, thechain rule implies[ n9 j=1[ y |(o ju +ui@) (x+0v)| dt |v].|@u(x+v) — @u(x)| dtlA(pu) ; (x-+tv) vjIATherefore, for such u,[,, |Outx+v)— ou (9) ax=f, ou(x+v)—ou(n)|axln< Lf Le |(6,0-+ 46) (x+4y)| dedelvlp nN< J, [,, EMG tei0) (x +09) adv< C(6 Mella)where C is a continuous function of ||u||, | y. Now for general u € W!! (UV), let u, > winW!+! (U) where uz, € C* (U) . Then for |v| < dist (spt() ,US) ,[,, |outx+v)— du(s)] ax[, (outst v) — ou(x)]a= lim I |ouy (x+v) — bug (x)] dxk> 00<i c(¢, )<= jim 9, \lMelli ay ) IVI= C(6.llulli aw) IvThis proves the lemma.