44.2. TRACE AND INTERPOLATION SPACES 1481
It is routine to verify from 44.2.33 that
w(m) (t) = (m−1)!(−1)m u( 1
t
)tm . (44.2.35)
For example, consider the case where m = 2.(∫∞
t
(1− t
τ
)u(
1τ
)dτ
τ
)′′=
(0+
∫∞
t
(−1
τ
)u(
1τ
)dτ
τ
)′=
1t2 u(
1t
).
Also from 44.2.33, it follows that trace(w) = a. It remains to verify w ∈ V m. From44.2.35, (∫
∞
0
(tθ+m−1
∣∣∣∣∣∣w(m) (t)∣∣∣∣∣∣
A1
)p dtt
)1/p
=
Cm
(∫∞
0
(tθ−1
∣∣∣∣∣∣∣∣u(1t
)∣∣∣∣∣∣∣∣A1
)pdtt
)1/p
=Cm
(∫∞
0
(t1−θ ||u(t)||A1
)p dtt
)1/p
≤Cm
(∫∞
0
(t−θ J (t,u(t))
)p dtt
)1/p
< ∞. (44.2.36)
It remains to consider(∫
∞
0
(tθ ||w(t)||A0
)pdtt
)1/p. From 44.2.34,
(∫∞
0
(tθ ||w(t)||A0
)p dtt
)1/p
=
(∫∞
0
(tθ
∣∣∣∣∣∣∣∣∫ 1
0(1− τ)m−1 u
(τ
t
) dτ
τ
∣∣∣∣∣∣∣∣A0
)pdtt
)1/p
=
(∫∞
0
(t−θ
∣∣∣∣∣∣∣∣∫ 1
0(1− τ)m−1 u(τt)
dτ
τ
∣∣∣∣∣∣∣∣A0
)pdtt
)1/p
≤∫ 1
0
(∫∞
0
(t−θ (1− τ)m−1 ||u(τt)||A0
)p dtt
)1/p dτ
τ
=∫ 1
0τ
θ (1− τ)m−1(∫
∞
0
(s−θ ||u(s)||A0
)p dss
)1/p dτ
τ
=
(∫ 1
0τ
θ−1 (1− τ)m−1 dτ
)(∫∞
0
(s−θ ||u(s)||A0
)p dss
)1/p
≤C(∫
∞
0
(s−θ ||u(s)||A0
)p dss
)1/p