45.4. FRACTIONAL ORDER SOBOLEV SPACES 1511
This proves the first part. Now consider the second. Let φ ∈C∞(Ω)
||Lφ ||m+σ ,p,Ω =
(||Lφ ||pm,p,Ω + ∑
|α|=m||Dα Lφ ||p
σ ,p,Ω
)1/p
=
(||Lφ ||pm,p,Ω + ∑
|α|=m||LDα
φ ||pT(W 1,p,Lp,p,1−σ)
)1/p
=
(||Lφ ||pm,p,Ω + ∑
|α|=m
[inf(∣∣∣∣t1−σ L fα
∣∣∣∣σ1
∣∣∣∣t1−σ L f ′α∣∣∣∣1−σ
2
)]p)1/p
(45.4.23)
whereinf(∣∣∣∣t1−σ L fα
∣∣∣∣σ1
∣∣∣∣t1−σ L f ′α∣∣∣∣1−σ
2
)=
inf(∣∣∣∣t1−σ L fα
∣∣∣∣σLp(0,∞; dt
t ;W 1,p(Ω))
∣∣∣∣t1−σ L f ′α∣∣∣∣1−σ
Lp(0,∞; dtt ;Lp(Ω))
),
fα (0)≡ limt→0 fα (t) = Dα φ in W 1,p (Ω)+Lp (Ω) , and the infimum is taken over all suchfunctions. Therefore, from 45.4.23, and letting ||L||1 denote the operator norm of L inW 1,p (Ω) and ||L||2 denote the operator norm of L in Lp (Ω) ,
||Lφ ||m+σ ,p,Ω
≤
(||Lφ ||pm,p,Ω + ∑
|α|=m
[inf(||L||σ1 ||L||
1−σ
2
∣∣∣∣t1−σ fα
∣∣∣∣σ1
∣∣∣∣t1−σ f ′α∣∣∣∣1−σ
2
)]p)1/p
≤
(||Lφ ||pm,p,Ω +
(||L||σ1 ||L||
1−σ
2
)p∑|α|=m
[inf(∣∣∣∣t1−σ fα
∣∣∣∣σ1
∣∣∣∣t1−σ f ′α∣∣∣∣1−σ
2
)]p)1/p
≤ C
(||φ ||pm,p,Ω + ∑
|α|=m
[||Dα
φ ||σ ,p,Ω
]p)1/p
=C ||φ ||m+σ ,p,Ω .
Since C∞(Ω)
is dense in all the Sobolev spaces, this inequality establishes the desiredresult.
Definition 45.4.6 Define for s≥ 0, W−s,p′ (Rn) to be the dual space of
W s,p (Rn) .
Here 1p +
1p′ = 1.
Note that in the case of m = 0 this is consistent with the Riesz representation theoremfor the Lp spaces.