1210 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

NowB≤C

(φ ,D′

)n−1/2||u′||WI ||w||WI .

Since u is bounded,

A ≤ C (φ ,u)∫ b

a

∫ 1

−1

∣∣∣∣∣∣D′ (t)−D′(

t− rn

)∣∣∣∣∣∣ ||w(t)||W drdt

≤ C (φ ,u)∫ b

a||w(t)||W n

∫ t+n−1

t−n−1

∣∣∣∣D′ (t)−D′ (s)∣∣∣∣dsdt

By Holder’s inequality, this is no larger than

C (φ ,u)(∫ b

a(n∫ t+n−1

t−n−1

∣∣∣∣D′ (t)−D′ (s)∣∣∣∣ds)2dt)1/2 ||w||WI

.

If t is a Lebesgue point,

n∫ t+n−1

t−n−1

∣∣∣∣D′ (t)−D′ (s)∣∣∣∣ds→ 0

and also

n∫ t+n−1

t−n−1

∣∣∣∣D′ (t)−D′ (s)∣∣∣∣ds≤ 4||D′||∞

so the dominated convergence theorem implies

∫ b

a(n∫ t+n−1

t−n−1

∣∣∣∣D′ (t)−D′ (s)∣∣∣∣ds)2dt→ 0.

Hence||Tnu||W ′I ≤

C(φ ,u,D′

)(n−1/2 +(

∫ b

a(n∫ t+n−1

t−n−1

∣∣∣∣D′ (t)−D′ (s)∣∣∣∣ds)2dt)1/2

)and so Tnu→ 0 for all u in the dense subset, C∞

0 (I;V ) .We have also the following simple corollary.

Corollary 34.6.2 In the situation of Proposition 34.6.1,∣∣∣∣(i∗D(u∗φ n))′− ((i∗Du)∗φ n)

′∣∣∣∣V ′I→ 0

where i is the inclusion map of V into W.

For f ∈ L1 (a,b;V ′) we define f ′ in the sense of V ′ valued distributions as follows. Forφ ∈C∞

0 (a,b) ,

f ′ (φ)≡−∫ b

af (t)φ

′ (t)dt.