44.1. DEFINITION AND BASIC THEORY OF TRACE SPACES 1475

Now || f (t)||A0+A1≤ || f (t)||A0

.

||a||A0+A1≤ tν || f (t)||A0+A1

t−ν +Cν t1−ν p′ || f ||W≤ tν || f (t)||A0

t−ν +Cν t1−ν p′ || f ||W

Therefore, recalling that ν p′ < 1, and integrating both sides from 0 to 1,

||a||A0+A1≤Cν || f ||W ≤Cν (||a||T + ε) .

To see this,

∫ 1

0tν || f (t)||A0

t−ν dt ≤(∫ 1

0

(tν || f (t)||A0

)pdt)1/p(∫ 1

0t−ν p′dt

)1/p′

≤ C || f ||W .

Since ε > 0 is arbitrary, this verifies the second inclusion and continuity of the inclusionmap completing the proof of the theorem.

The interpolation inequality, 44.1.7 is very significant. The next result concerns bound-ed linear transformations.

Theorem 44.1.10 Now suppose A0,A1 and B0, B1 are pairs of Banach spaces such that Aiembeds continuously into a topological vector space, X and Bi embeds continuously into atopological vector space, Y. Suppose also that L ∈L (A0,B0) and L ∈L (A1,B1) wherethe operator norm of L in these spaces is Ki, i = 0,1. Then

L ∈L (A0 +A1,B0 +B1) (44.1.11)

with||La||B0+B1

≤max(K0,K1) ||a||A0+A1(44.1.12)

andL ∈L (T (A0,A1, p,θ) ,T (B0,B1, p,θ)) (44.1.13)

and for K the operator norm,K ≤ K1−θ

0 Kθ1 . (44.1.14)

Proof: To verify 44.1.11, let a ∈ A0 +A1 and pick a0 ∈ A0 and a1 ∈ A1 such that

||a||A0+A1+ ε > ||a0||A0

+ ||a1||A1.

Then||L(a)||B0+B1

= ||La0 +La1||B0+B1≤ ||La0||B0

+ ||La1||B1

≤ K0 ||a0||A0+K1 ||a1||A1

≤max(K0,K1)(||a||A0+A1

+ ε

).

This establishes 44.1.12. Now consider the other assertions.