1494 CHAPTER 45. TRACES OF SOBOLEV SPACES
Let φ be a nonnegative decreasing infinitely differentiable function such that φ (0) = 1 andφ (t) = 0 for all t > 1. Then define
f (t)≡ φ (t)1t
∫ t
0G(τ)udτ.
It is easy to see that f (t) ∈ D(Λ) . In fact, changing variables as needed,
1h
(G(h)
∫ t
0G(τ)udτ−
∫ t
0G(τ)udτ
)
=1h
∫ t+h
hG(τ)udτ− 1
h
∫ t
0G(τ)udτ
=1h
∫ t+h
tG(τ)udτ− 1
h
∫ h
0G(τ)udτ
which converges to G(t)u−u and so
Λ
∫ t
0G(τ)udτ = G(t)u−u. (45.3.12)
Thus ∫∞
0t pθ ||Λ f ||pA1
dtt≤
∫∞
0t pθ
∣∣∣∣∣∣∣∣G(t)u−ut
∣∣∣∣∣∣∣∣pA1
dtt
≤ ||u||pT0
Next it is necessary to consider ∫∞
0t pθ∣∣∣∣ f ′∣∣∣∣pA1
dtt.
f ′ (t) = φ′ (t)
1t
∫ t
0G(τ)udτ+
φ (t)(− 1
t2
∫ t
0G(τ)udτ +
1t
G(t)u)
= φ′ (t)
1t
∫ t
0G(τ)udτ +φ (t)
(1t2
∫ t
0(G(t)u−G(τ)u)dτ
)and so there is a constant C depending on φ and the uniform bound on ||G(t)|| such that
∣∣∣∣ f ′ (t)∣∣∣∣A1≤ CX[0,1] (t)
(||u||A1
+1t2
∫ t
0||G(t− τ)u−u||dτ
)= CX[0,1] (t)
(||u||A1
+1t2
∫ t
0||G(τ)u−u||dτ
)