1220 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

=k

∑i=1

1ti− ti−1

∫ ti

ti−1

un (t)dsX[ti−1,ti) (t)−k

∑i=1

1ti− ti−1

∫ ti

ti−1

un (s)dsX[ti−1,ti) (t)

=k

∑i=1

1ti− ti−1

∫ ti

ti−1

(un (t)−un (s))dsX[ti−1,ti) (t) .

It follows from Jensen’s inequality that

||un (t)−un (t)||pW

=k

∑i=1

∣∣∣∣∣∣∣∣ 1ti− ti−1

∫ ti

ti−1

(un (t)−un (s))ds∣∣∣∣∣∣∣∣p

WX[ti−1,ti) (t)

≤k

∑i=1

1ti− ti−1

∫ ti

ti−1

||un (t)−un (s)||pW dsX[ti−1,ti) (t)

and so ∫ b

a||(un (t)−un (s))||pW ds

≤∫ b

a

k

∑i=1

1ti− ti−1

∫ ti

ti−1

||un (t)−un (s)||pW dsX[ti−1,ti) (t)dt

=k

∑i=1

1ti− ti−1

∫ ti

ti−1

∫ ti

ti−1

||un (t)−un (s)||pW dsdt. (34.7.59)

From Theorem 34.7.2 if ε > 0, there exists Cε such that

||un (t)−un (s)||pW ≤ ε ||un (t)−un (s)||pE +Cε ||un (t)−un (s)||pX

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

This is substituted in to 34.7.59 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 |t− s|p/q)

dsdt

=k

∑i=1

2pε

∫ ti

ti−1

||un (t)||pW +Cε

ti− ti−1

∫ ti

ti−1

∫ ti

ti−1

|t− s|p/q dsdt

≤ 2pε

∫ b

a||un (t)||p dt +Cε

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

k

∑i=1

1(ti− ti−1)

(ti− ti−1)p/q (ti− ti−1)

2

≤ 2pεRp +Cε

k

∑i=1

(ti− ti−1)1+p/q = 2p

εRp +Cε k(

b−ak

)1+p/q

.