1510 CHAPTER 45. TRACES OF SOBOLEV SPACES

Then Lk is continuous as a map from W 1,p (V ) to W 1,p (U) and as a map from Lp (V ) toLp (U) and therefore, it follows that Lk is continuous as a map from W σ ,p (V ) to W σ ,p (U) .Therefore,

||Lk (v)||σ ,p,U ≤Ck ||v||σ ,p,U

and so ∣∣∣∣D j (h∗u)∣∣∣∣

σ ,p,U ≤ ∑k

∣∣∣∣Lk(u,k)∣∣∣∣

σ ,p,U

≤ ∑k

Ck ||Dku||σ ,p,V

≤ C

(∑k||Dku||p

σ ,p,V

)1/p

.

Therefore, it follows that

||h∗u||1+σ ,p,U ≤

[||h∗u||p1,p,U +∑

jCp

∑k||Dku||p

σ ,p,V

]1/p

≤ C

[||u||p1,p,V +∑

k||Dku||p

σ ,p,V

]1/p

=C ||u||1+σ ,p,V .

The general case is similar except for the use of a more complicated linear operator in placeof Lk. This proves the theorem.

It is interesting to prove this theorem using Theorem 45.3.13 and the intrinsic norm.Now we prove an important interpolation inequality for Sobolev spaces.

Theorem 45.4.5 Let Ω be an open set in Rn which has the segment property and let f ∈W m+1,p (Ω) and σ ∈ (0,1) . Then for some constant, C, independent of f ,

|| f ||m+σ ,p,Ω ≤C || f ||1−σ

m+1,p,Ω || f ||σ

m,p,Ω .

Also, if L ∈L (W m,p (Ω) ,W m,p (Ω)) for all m = 0,1, · · · , and L◦Dα = Dα ◦L on C∞(Ω),

then L ∈L (W m+σ ,p (Ω) ,W m+σ ,p (Ω)) for any m = 0,1, · · · .

Proof: Recall from above, W 1−θ ,p (Ω) ≡ T(W 1,p (Ω) ,Lp (Ω) , p,θ

). Therefore, from

Theorem 44.1.9, if f ∈W 1,p (Ω) ,

|| f ||1−θ ,p,Ω ≤ K || f ||θ1,p,Ω || f ||1−θ

0,p,Ω

Therefore,

|| f ||m+σ ,p,Ω ≤

(|| f ||pm,p,Ω + ∑

|α|=mK(||Dα f ||1−σ

1,p,Ω ||Dα f ||σ0,p,Ω

)p)1/p

≤ C[|| f ||pm,p,Ω +

(|| f ||1−σ

m+1,p,Ω || f ||σ

m,p,Ω

)p]1/p

≤ C[(|| f ||1−σ

m+1,p,Ω || f ||σ

m,p,Ω

)p+(|| f ||1−σ

m+1,p,Ω || f ||σ

m,p,Ω

)p]1/p

≤ C || f ||1−σ

m+1,p,Ω || f ||σ

m,p,Ω .