1476 CHAPTER 44. TRACE SPACES

Let a ∈ T (A0,A1, p,θ) and pick f ∈W (A0,A1, p,θ) such that γ f = a and

||a||T (A0,A1,p,θ)+ ε >∣∣∣∣∣∣tθ f

∣∣∣∣∣∣1−θ

Lp(0,∞, dtt ,A0)

∣∣∣∣∣∣tθ f ′∣∣∣∣∣∣θ

Lp(0,∞, dtt ,A1)

.

Then consider L f . Since L is continuous on A0 +A1,

L f (0) = La

and L f ∈W (B0,B1, p,θ) . Therefore, by Theorem 44.1.9,

||La||T (B0,B1,p,θ) ≤∣∣∣∣∣∣tθ L f

∣∣∣∣∣∣1−θ

Lp(0,∞, dtt ,B0)

∣∣∣∣∣∣tθ L f ′∣∣∣∣∣∣θ

Lp(0,∞, dtt ,B1)

≤ K1−θ

0 Kθ1

∣∣∣∣∣∣tθ f∣∣∣∣∣∣1−θ

Lp(0,∞, dtt ,A0)

∣∣∣∣∣∣tθ f ′∣∣∣∣∣∣θ

Lp(0,∞, dtt ,A1)

≤ K1−θ

0 Kθ1

(||a||T (A0,A1,p,θ)+ ε

).

and since ε > 0 is arbitrary, this proves the theorem.

44.2 Trace And Interpolation SpacesTrace spaces are equivalent to interpolation spaces. In showing this, a more general sort oftrace space than that presented earlier will be used.

Definition 44.2.1 Define for m a positive integer, V m = V m (A0,A1, p,θ) to be the set offunctions, u such that

t→ tθ u(t) ∈ Lp(

0,∞,dtt

;A0

)(44.2.15)

and

t→ tθ+m−1u(m) (t) ∈ Lp(

0,∞,dtt

;A1

). (44.2.16)

V m is a Banach space with the norm

||u||V m ≡max(∣∣∣∣∣∣tθ u(t)

∣∣∣∣∣∣Lp(0,∞, dt

t ;A0),∣∣∣∣∣∣tθ+m−1u(m) (t)

∣∣∣∣∣∣Lp(0,∞, dt

t ;A1)

).

Thus V m equals W in the case when m = 1. More generally, as in [16] different expo-nents are used for the two Lp spaces, p0 in place of p for the space corresponding to A0 andp1 in place of p for the space corresponding to A1.

Definition 44.2.2 Denote by T m (A0,A1, p,θ) the set of all a ∈ A0 +A1 such that for someu ∈V m,

a = limt→0+

u(t)≡ trace(u) , (44.2.17)

the limit holding in A0 +A1. For the norm

||a||T m ≡ inf{||u||V m : trace(u) = a} . (44.2.18)