1276 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

the Ascoli Arzela theorem, there exists a subsequence of this, still denoted by fk such that∑|α|≤m ||Dα ( fl− fk)||∞ < δ where δ satisfies

0 < δ < min(

ε

2,(

ε

8

)(ε

8M

)β/(λ−β )). (37.1.14)

Therefore, sup|α|=m ρβ (Dα ( fk− fl))≥ ε−δ for all k ̸= l. It follows that there exist pairs

of points and a multi index, α with |α|= m, {xkl ,ykl ,α} such that

ε−δ

2<|(Dα fk−Dα fl)(xkl)− ((Dα fk−Dα fl)(ykl))|

|xkl−ykl |β≤ 2M |xkl−ykl |λ−β (37.1.15)

and so considering the ends of the above inequality,(ε−δ

4M

)1/(λ−β )

< |xkl−ykl | .

Now also, since ∑|α|≤m ||Dα ( fl− fk)||∞ < δ , it follows from the first inequality in 37.1.15that

ε−δ

2<

2δ(ε−δ

4M

)β/(λ−β ).

Since δ < ε/2, this impliesε

4<

2δ(ε

8M

)β/(λ−β )

and so (ε

8

)(ε

8M

)β/(λ−β )< δ

contrary to 37.1.14. This proves the lemma.

Corollary 37.1.7 Let p > n,U and rU be as in Theorem 37.1.3 and let m be a nonnegativeinteger. Then rU : W m+1,p (Rn)→Cm,λ

(U)

is continuous as a map into Cm,λ(U)

for allλ ∈ [0,1− n

p ] and rU is compact if λ < 1− np .

Proof: Suppose uk→ 0 in W m+1,p (Rn) . Then from 37.1.13, if λ ≤ 1− np and |α|= m

ρλ (Dα uk)≤C ||Dα uk||1,p diam(U)1− n

p−λ .

Therefore, ρλ (Dα uk)→ 0. From Theorem 37.1.3 it follows that for |α| ≤ m,

||Dα uk||∞→ 0

and so ||uk||m,λ → 0. This proves the claim about continuity. The claim about compactnessfor λ < 1− n

p follows from Lemma 37.1.6 and this.

(Bounded in W m,p (Rn)rU→ Bounded in Cm,1− n

p(U) id→ Compact in Cm,λ

(U).)

It is just as important to consider the case where p < n. To do this case the followinglemma due to Gagliardo [53] will be of interest. See also [1].