Kenneth Kuttler

1310 CHAPTER 38. SOBOLEV SPACES BASED ON L2

Therefore, where C (n,m) is the largest of the multinomial coefficients obtained in the ex-pansion, (

1+n

∑j=1

z2j

Therefore,

|||u|||2m+s

≤ C∫ (

1+ |z|2)m|Fu(z)|2 dz+M (s)

∫|Fu(z)|2 ∑

|α|=m|zα |2 |z|2s dz

≤ C∫ (

1+ |z|2)m+s

|Fu(z)|2 dz+M (s)∫|Fu(z)|2

(1+ |z|2

)m|z|2s dz

≤ C∫ (

1+ |z|2)m+s

|Fu(z)|2 dz =C ||u||Hm+s(Rn) .

It remains to show the other inequality. From 38.3.11,∫ ∫|Dα u(x)−Dα u(y)|2 |x−y|−n−2s dxdy

≥ m(s)∫|FDα u(z)|2 |z|2s dz

= m(s)∫|Fu(z)|2 |zα |2 |z|2s dz.

No reference was made to |α| = m and so this establishes the bottom half of 38.3.10.Therefore, from 38.3.12,

|||u|||2m+s

≥ C∫ (

1+ |z|2)m|Fu(z)|2 dz+m(s)

∫|Fu(z)|2 ∑

|α|=m|zα |2 |z|2s dz

≥ C∫ (

1+ |z|2)m|Fu(z)|2 dz+C

∫|Fu(z)|2

(1+ |z|2

)m|z|2s dz

= C∫ (

1+ |z|2)m(

1+ |z|2s)|Fu(z)|2 dz

≥ C∫ (

1+ |z|2)m(

1+ |z|2)s|Fu(z)|2 dz

= C∫ (

1+ |z|2)m+s

|Fu(z)|2 dz = ||u||Hm+s(Rn) .

This proves the theorem.With the above intrinsic norm, it becomes possible to prove the following version of

Theorem 38.2.7.

Lemma 38.3.2 Let h : Rn → Rn be one to one and onto. Also suppose that Dα h andDα(h−1)

exist and are Lipschitz continuous if |α| ≤ m for m a positive integer. Then

h∗ : Hm+s (Rn)→ Hm+s (Rn)

1310 CHAPTER 38. SOBOLEV SPACES BASED ON L?Therefore, where C (n,m) is the largest of the multinomial coefficients obtained in the ex-pansion,n m14 VG] .j=l2[Neel lintm< Cc (1+ 12’) [Fu(2)[Pdz+M (s) [ \Fu (a)? Y |2*/? ||? az|@|=m< cf (1+ ie?) rua) Pde +m (9) [ [ru (e)l? (1+ leP)” JalaTherefore,m+s< of (1+ (a?)" Futa)tde=C ill pmsenIt remains to show the other inequality. From 38.3.11,[ [ }%u(x) Duly)? jx yl" aryIVm(s) | \FD%u(2)[?|al* dzm(s) | \Fu(a)|? a")? 2)? dzNo reference was made to |a| = m and so this establishes the bottom half of 38.3.10.Therefore, from 38.3.12,I|==2241 Nines2\" 2s> C/ (14 |z?) |Fu(2)P?dz+m(s) | |Fu(z > 2 |? \al7° dz|o|=m2\™" 2\""\_ 12s> C (1+2 |Fu(z \Pdz+C ‘|Fu(z *(14|z| |z\" dz2= cf(i +|z2\" 2> C (1+ (1+ (2? "|Fu(z)| dz))y"( (1+) |Fu(2)|?dzy(ye2 m+- i” b+ |a?)" [Fu (2) dz = ltl ymsscn)This proves the theorem.With the above intrinsic norm, it becomes possible to prove the following version ofTheorem 38.2.7.Lemma 38.3.2 Let h: R” — R"” be one to one and onto. Also suppose that D“h andD* (h- ') exist and are Lipschitz continuous if |o| <_m for m a positive integer. Thenh* » ints (R”) _ H™t (R”)