1234 CHAPTER 35. WEAK DERIVATIVES
Proof: T (Dφ) = 0 for all φ ∈C∞c (a,b) from the definition of DT = 0. Let
φ 0 ∈C∞c (a,b) ,
∫ b
aφ 0 (x)dx = 1,
and let
ψφ (x) =∫ x
a[φ (t)−
(∫ b
aφ (y)dy
)φ 0 (t)]dt
for φ ∈C∞c (a,b). Thus ψφ ∈C∞
c (a,b) and
Dψφ = φ −(∫ b
aφ (y)dy
)φ 0.
Therefore,
φ = Dψφ +
(∫ b
aφ (y)dy
)φ 0
and so
T (φ) = T (Dψφ )+
(∫ b
aφ (y)dy
)T (φ 0) =
∫ b
aT (φ 0)φ (y)dy.
Let C = T φ 0. This proves the lemma.Proof of Theorem 35.2.2 Since f and D f are both in L1 (a,b),
D f (φ)−∫ b
aD f (x)φ (x)dx = 0.
Consider
f (·)−∫ (·)
aD f (t)dt
and let φ ∈C∞c (a,b).
D(
f (·)−∫ (·)
aD f (t)dt
)(φ)
≡−∫ b
af (x)φ
′ (x)dx+∫ b
a
(∫ x
aD f (t)dt
)φ′ (x)dx
= D f (φ)+∫ b
a
∫ b
tD f (t)φ
′ (x)dxdt
= D f (φ)−∫ b
aD f (t)φ (t)dt = 0.
By Lemma 35.2.3, there exists a constant, C, such that(f (·)−
∫ (·)
aD f (t)dt
)(φ) =
∫ b
aCφ (x)dx