40.1. A FUNDAMENTAL INEQUALITY 1345
=Cn sup
∣∣∣∣∣∣∣∫Rn
ti (Fφ)(
1+ |t|2)1/2
(1+ |t|2
)1/2 (F f )dt
∣∣∣∣∣∣∣ : ||φ ||1,2 ≤ 1
=Cn
∫ |F f |2 t2i(
1+ |t|2)dt
1/2
(40.1.5)
Also,
|| f ||−1,2 ≡ sup{∣∣∣∣∫Rn
φ f dx∣∣∣∣ : ||φ ||1,2 ≤ 1
}
=Cn sup{∣∣∣∣∫Rn
(Fφ)(F f)
dx∣∣∣∣ : ||φ ||1,2 ≤ 1
}
=Cn sup
∣∣∣∣∣∣∣∫Rn
Fφ
(1+ |t|2
)1/2
(1+ |t|2
)1/2 (F f )dt
∣∣∣∣∣∣∣ : ||φ ||1,2 ≤ 1
=Cn
∫Rn
|F f |2(1+ |t|2
)dt
1/2
This along with 40.1.5 yields the conclusion of the lemma because
n
∑i=1
∣∣∣∣∣∣∣∣ ∂ f∂xi
∣∣∣∣∣∣∣∣2−1,2
+ || f ||2−1,2 =Cn
∫Rn|F f |2 dx =Cn || f ||20,2 .
Now consider Theorem 40.1.1. First note that by Lemma 40.1.2 and U− defined there,Lemma 40.1.3 implies that for f extended as in Lemma 40.1.2,
|| f ||0,2,U− ≤ || f ||0,2,Rn =Cn
(|| f ||−1,2,Rn +
n
∑i=1
∣∣∣∣∣∣∣∣ ∂ f∂xi
∣∣∣∣∣∣∣∣−1,2,Rn
)
≤Cgn
(|| f ||−1,2,U− +
n
∑i=1
∣∣∣∣∣∣∣∣ ∂ f∂xi
∣∣∣∣∣∣∣∣−1,2,U−
). (40.1.6)
Let Ω be a bounded open set having Lipschitz boundary which lies locally on one sideof its boundary. Let {Qi}p
i=0 be cubes of the sort used in the proof of the divergencetheorem such that Q0 ⊆Ω and the other cubes cover the boundary of Ω. Let {ψ i} be a C∞
partition of unity with spt(ψ i)⊆Qi and let f ∈ L2 (Ω) . Then for φ ∈C∞c (Ω) and ψ one of
these functions in the partition of unity,∣∣∣∣∣∣∣∣∂ ( f ψ)
∂xi
∣∣∣∣∣∣∣∣−1,2,Ω
≤ sup||φ ||1,2≤1
∣∣∣∣∫Ω
f∂
∂xi(ψφ)dx
∣∣∣∣+ sup||φ ||1,2≤1
∣∣∣∣∫Ω
f φ∂ψ
∂xidx∣∣∣∣