1728 CHAPTER 54. FUNCTIONAL ANALYSIS APPLICATIONS

Proof: The first part is done in the above theorem. Let S be a bounded set and letη > 0. Then let ε > 0 be small enough that for all x ∈ S,ε ||x|| < η/4. Let

{(−A)−1 xn

}be a η/(2+2C (ε,α)) net for (−A)−1 (S) . Then if (−A)−α x ∈ (−A)−α S, there exists xnsuch that ∣∣∣∣∣∣(−A)−1 xn− (−A)−1 x

∣∣∣∣∣∣< η

2+2C (ε,α).

Then∣∣∣∣(−A)−α xn− (−A)−α x∣∣∣∣ ≤ ε ||xn− x||+C (ε,α)

∣∣∣∣∣∣(−A)−1 xn− (−A)−1 x∣∣∣∣∣∣

<η

2+

η

2= η

showing (−A)−α (S) has a η net. Thus (−A)−α is compact. This proves the corollary.The next proposition gives a general interpolation inequality.

Proposition 54.3.21 Let 0 < α < β and let

γ = θβ +(1−θ)α, θ ∈ (0,1) .

Then there exists a constant, C such that for all x ∈ D((−A)β

),

∣∣∣∣(−A)γ x∣∣∣∣≤C

∣∣∣∣∣∣(−A)β x∣∣∣∣∣∣θ ∣∣∣∣(−A)α x

∣∣∣∣1−θ.

Proof: This is an exercise in using 54.3.22. Letting x ∈ D((−A)β

),

(−A)γ x = (−A)θ (−A)−θ (−A)γ x

Therefore, letting C denote a generic constant, it follows since (−A)θ is closed,

Γ(θ)∣∣∣∣(−A)γ x

∣∣∣∣= ∣∣∣∣∣∣∣∣∫ ∞

0tθ−1 (−A)θ S (t)(−A)γ xdt

∣∣∣∣∣∣∣∣≤

∫η

0tθ−1

∣∣∣∣∣∣(−A)θ (−A)γ−β S (t)(−A)β x∣∣∣∣∣∣dt

+∫

∞

η

tθ−1∣∣∣∣∣∣(−A)θ (−A)γ−α S (t)(−A)α x

∣∣∣∣∣∣dt

≤ C∫

η

0tθ−1t−θ tβ−γ dt

∣∣∣∣∣∣(−A)β x∣∣∣∣∣∣+C

∫∞

η

tθ−1t−θ tα−γ dt∣∣∣∣(−A)α x

∣∣∣∣= C

(ηβ−γ

β − γ

∣∣∣∣∣∣(−A)β x∣∣∣∣∣∣+ η−(γ−α)

γ−α

∣∣∣∣(−A)α x∣∣∣∣)