1482 CHAPTER 44. TRACE SPACES
≤C(∫
∞
0
(t−θ J (t,u(t))
)pdt)1/p
< ∞. (44.2.37)
It follows that||w||V m ≡
max
((∫∞
0
(tθ ||w(t)||A0
)p dtt
)1/p
,
(∫∞
0
(tθ+m−1
∣∣∣∣∣∣w(m) (t)∣∣∣∣∣∣
A1
)p dtt
)1/p)
≤C(∫
∞
0
(t−θ J (t,u(t))
)pdt)1/p
< ∞
which shows that a ∈ T m (A0,A1,θ , p) . Taking the infimum,
||a||T m ≤C ||a||θ ,p,J .
This together with 44.2.32 proves the theorem.By Theorem 44.2.3 and Theorem 43.8.6, we obtain the following important corollary
describing the dual space of a trace space.
Corollary 44.2.4 Let A0∩A1 be dense in Ai for i = 0,1 and suppose that Ai is reflexive fori = 0,1. Then for ∞ > p≥ 1,
T m (A0,A1,θ , p)′ = T m (A′1,A′0,1−θ , p′)
1482 CHAPTER 44. TRACE SPACES<e([(re@uin))" ar) 1 oe snapIt follows that|W] lym =_ (Uf (+? lb (llaa)” 7) . UL (im Iara) .) 7) ')\/p<C (f (1-24 (e.u(n))" ar) <0which shows that a € T™ (Ag,A1, 9, p). Taking the infimum,Hlallrm <C|lallo psThis together with 44.2.32 proves the theorem.By Theorem 44.2.3 and Theorem 43.8.6, we obtain the following important corollarydescribing the dual space of a trace space.Corollary 44.2.4 Let Aj A, be dense in A; for i= 0,1 and suppose that A; is reflexive fori=0,1. Then foro > p> 1,T™ (Ao,A1,9,p)’ =T" (Ai, Ao, 1— 0, p')