43.5. SOME IMBEDDING THEOREMS 1447

≤ 2p−1ε (||un (t)||p + ||un (s)||p)+Cε

∣∣∣∣∣∣∣∣∫ t

su′n (r)dr

∣∣∣∣∣∣∣∣pX

≤ 2p−1ε (||un (t)||p + ||un (s)||p)+Cε

(∫ t

s

∣∣∣∣u′n (r)∣∣∣∣X dr)p

≤ 2p−1ε (||un (t)||p + ||un (s)||p)

+Cε

((∫ t

s

∣∣∣∣u′n (r)∣∣∣∣qX dr)1/q

|t− s|1/q′)p

= 2p−1ε (||un (t)||p + ||un (s)||p)+Cε Rp/q |t− s|p/q′ .

This is substituted in to 43.5.25 to obtain∫ b

a||(un (t)−un (s))||pW ds≤

k

∑i=1

1ti− ti−1

∫ ti

ti−1

∫ ti

ti−1

(2p−1

ε (||un (t)||p + ||un (s)||p)

+Cε Rp/q |t− s|p/q′)

dsdt

=k

∑i=1

2pε

∫ ti

ti−1

||un (t)||pW +Cε Rp/q 1ti− ti−1

∫ ti

ti−1

∫ ti

ti−1

|t− s|p/q′ dsdt

= 2pε

∫ b

a||un (t)||p dt +Cε Rp/q

k

∑i=1

1(ti− ti−1)

(ti− ti−1)p/q′

∫ ti

ti−1

∫ ti

ti−1

dsdt

= 2pε

∫ b

a||un (t)||p dt +Cε Rp/q

k

∑i=1

1(ti− ti−1)

(ti− ti−1)p/q′ (ti− ti−1)

2

≤ 2pεRp +Cε Rp/q

k

∑i=1

(ti− ti−1)1+p/q′ = 2p

εRp +Cε Rp/qk(

b−ak

)1+p/q′

.

Taking ε so small that 2pεRp < η p/8p and then choosing k sufficiently large, it follows

||un−un||Lp([a,b];W ) <η

4.

Now use compactness of the embedding of E into W to obtain a subsequence such that{un} is Cauchy in Lp (a,b;W ) and use this to contradict 43.5.24. The details follow.

Suppose un (t) = ∑ki=1 un

i X[ti−1,ti) (t) . Thus

||un (t)||E =k

∑i=1||un

i ||E X[ti−1,ti) (t)

and so

R≥∫ b

a||un (t)||pE dt =

Tk

k

∑i=1||un

i ||pE