Master Dissertation

Now choose a monotonous function χ0 ∈ C∞ (R) such that
⎧
⎪⎨ 0, t ≤ 0,
χ0(t) = s ∈ [0, 1),
⎪⎩
1,
0 < t < 1,
t ≥ 1.
The following theorem gives us a retarded distribution.
Theorem 7.7. The limit
exists.
v · x def
lim χ0( )d(x) = θ(v · x)d(x) (7.11)
δ→0 δ
In order to prove this we need the following lemma.
Lemma 7.8. Let a1 > 1, then
ψ0( x
)d(x) :=
δ
uniformly in a ≥ a1.
χ0(a
v · x v · x
) − χ0(
δ δ )
d(x) → 0 for δ → 0
Proof. Let U be a neighborhood of Γ + (0) ∪ Γ − (0). Choose a function
ψ1 ∈ C ∞ (R m ) such that
Then, by (5.20)
⎧
⎪⎨
ψ1(x) =
⎪⎩
0, Rm \ U.
〈ψ0( x
δ
1, x ∈ Γ + (0) ∪ Γ − (0),
s ∈ (0, 1], U \ Γ + (0) ∪ Γ − (0),
x x
)d(x), φ(x)〉 = 〈ψ0( )d(x), ψ1(
δ δ )φ(x)〉
= 〈φ(x)d(x), ψ0( x
δ
for any φ ∈ S(R m ), since ψ0ψ1 ∈ C ∞ 0 (Rm ).
By (7.1)
dδ def
= ρ(δ)δ m d(δx) → d0
for δ → 0. Further
in C ∞ (R m ) since
φ(δx) → φ(0) for δ → 0
D α (φ(δx) − φ(0)) = δ |α| (D α φ)(δx) − D α φ(0).
Therefore by [12] Theorem 6.1.8
x
)ψ1( )〉, (7.12)
δ
φ(δx)dδ → φ(0)d0 for δ → 0. (7.13)
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