44.1. DEFINITION AND BASIC THEORY OF TRACE SPACES 1473

Proof: Define a mapping, ψ : W/Z→ T by

ψ ([ f ])≡ γ f .

Then ψ is one to one and onto. Also

||[ f ]|| ≡ inf{|| f +g|| : g ∈ Z}= inf{||h||W : γh = γ f}= ||γ ( f )||T .

Therefore, the Banach space, W/Z and T are isometric and so T must be a Banach spacesince W/Z is.

The following is an important interpolation inequality.

Theorem 44.1.9 If a ∈ T, then

||a||T = inf{∣∣∣∣∣∣tθ f

∣∣∣∣∣∣1−θ

Lp(0,∞, dtt ;A0)

∣∣∣∣∣∣tθ f ′∣∣∣∣∣∣θ

Lp(0,∞, dtt ;A1)

}(44.1.6)

where the infimum is taken over all f ∈W such that a = f (0) . Also, if a ∈ A0 ∩A1, thena ∈ T and

||a||T ≤ K ||a||1−θ

A1||a||θA0

(44.1.7)

for some constant K. Also

A0∩A1 ⊆ T (A0,A1, p,θ)⊆ A0 +A1 (44.1.8)

and the inclusion maps are continuous.

Proof: First suppose f (0) = a where f ∈W . Then letting fλ (t) ≡ f (λ t) , it followsthat fλ (0) = a also and so

||a||T ≤ max(∣∣∣∣∣∣tθ fλ

∣∣∣∣∣∣Lp(0,∞, dt

t ;A0),∣∣∣∣∣∣tθ ( fλ )

′∣∣∣∣∣∣

Lp(0,∞, dtt ;A1)

)= max

(λ−θ

∣∣∣∣∣∣tθ f∣∣∣∣∣∣

Lp(0,∞, dtt ;A0)

,λ 1−θ

∣∣∣∣∣∣tθ f ′∣∣∣∣∣∣

Lp(0,∞, dtt ;A1)

)≡ max

(λ−θ R,λ 1−θ S

).

Now choose λ = R/S to obtain

||a||T ≤ R1−θ Sθ =∣∣∣∣∣∣tθ f

∣∣∣∣∣∣1−θ

Lp(0,∞, dtt ;A0)

∣∣∣∣∣∣tθ f ′∣∣∣∣∣∣θ

Lp(0,∞, dtt ;A1)

.

Thus

||a||T ≤ inf{∣∣∣∣∣∣tθ f

∣∣∣∣∣∣1−θ

Lp(0,∞, dtt ;A0)

∣∣∣∣∣∣tθ f ′∣∣∣∣∣∣θ

Lp(0,∞, dtt ;A1)

}.

Next choose f ∈W such that f (0) = a and || f ||W ≈ ||a||T . More precisely, pick f ∈Wsuch that f (0) = a and ||a||T >−ε + || f ||W . Also let

R≡∣∣∣∣∣∣tθ f

∣∣∣∣∣∣Lp(0,∞, dt

t ;A0),S≡

∣∣∣∣∣∣tθ f ′∣∣∣∣∣∣

Lp(0,∞, dtt ;A1)

.