Chapter 44

Trace Spaces44.1 Definition And Basic Theory Of Trace Spaces

Another approach to these sorts of problems is to use trace spaces. This allows the con-sideration of fractional order Sobolev spaces. In so far as the subject of Sobolev spaces isconcerned, I will present this material in a manner which is essentially independent of theprevious material on interpolation spaces.

As in the case of interpolation spaces, suppose A0 and A1 are two Banach spaces whichare continuously embedded in some topological vector space, X .

Definition 44.1.1 Define a norm on A0 +A1 as follows.

||a||A0+A1≡ inf

{||a0||A0

+ ||a1||A1: a0 +a1 = a

}(44.1.1)

Lemma 44.1.2 A0 +A1 with the norm just described is a Banach space.

Proof: This was already explained in the treatment of the K method of interpolation. Itis just K (1,a) .

Definition 44.1.3 Take f ′ in the sense of distributions for any

f ∈ L1loc (0,∞;A0 +A1)

as follows.

f ′ (φ)≡∫

∞

0− f (t)φ

′ (t)dt

whenever φ ∈ C∞c (0,∞) . Define a Banach space, W (A0,A1, p,θ) = W where p ≥ 1,θ ∈

(0,1). Let

|| f ||W ≡max(∣∣∣∣∣∣tθ f

∣∣∣∣∣∣Lp(0,∞, dt

t ;A0),∣∣∣∣∣∣tθ f ′

∣∣∣∣∣∣Lp(0,∞, dt

t ;A1)

)(44.1.2)

and let W consist of f ∈ L1loc (0,∞;A0 +A1) such that || f ||W < ∞.

Note that to be in W, f (t) ∈ A0 and f ′ (t) ∈ A1.

Lemma 44.1.4 If f ∈W, then

Trace( f )≡ f (0)≡ limt→0

f (t)

exists in A0 + A1. Also Z ≡ { f ∈W : f (0) = 0} is a closed subspace of W. In additionto this, for every f ∈ W and ε > 0 there exists a g ∈ W such that || f −g||W < ε andg ∈C∞ (0,∞;A0) while g′ ∈C∞ (0,∞;A1).

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