38.3. AN INTRINSIC NORM 1313

Since this is true for all v ∈ Hm+s (Rn) , it follows that

||h∗u||Hm+s(U) ≤C ||u||Hm+s(V ) .

With harder work, you don’t need to have h,h−1 defined on all of Rn but I don’t feellike including the details so this lemma will suffice.

Another interesting application of the intrinsic norm is the following.

Lemma 38.3.4 Let φ ∈ Cm,1 (Rn) and suppose spt(φ) is compact. Then there exists aconstant, Cφ such that whenever u ∈ Hm+s (Rn) ,

||φu||Hm+s(Rn) ≤Cφ ||u||Hm+s(Rn) .

Proof: It is a routine exercise in the product rule to verify that

||φu||Hm(Rn) ≤Cφ ||u||Hm(Rn) .

It only remains to consider the term involving the integral. A typical term is∫ ∫|Dα

φu(x)−Dαφu(y)|2 |x−y|−n−2s dxdy.

This is a finite sum of terms of the form∫ ∫ ∣∣∣Dγφ (x)Dβ u(x)−Dγ

φ (y)Dβ u(y)∣∣∣2 |x−y|−n−2s dxdy

where |γ| and |β | ≤ m.

≤ 2∫ ∫

|Dγφ (x)|2

∣∣∣Dβ u(x)−Dβ u(y)∣∣∣2 |x−y|−n−2s dxdy

+2∫ ∫ ∣∣∣Dβ u(y)

∣∣∣2 |Dγφ (x)−Dγ

φ (y)|2 |x−y|−n−2s dxdy

By 38.3.10 and the Lipschitz continuity of all the derivatives of φ , this is dominated by

CM (s)∫|Fu(z)|2

∣∣∣zβ

∣∣∣2 |z|2s dz

+K∫ ∫ ∣∣∣Dβ u(y)

∣∣∣2 |x−y|2 |x−y|−n−2s dxdy

= CM (s)∫|Fu(z)|2

∣∣∣zβ

∣∣∣2 |z|2s dz

+K∫ ∣∣∣Dβ u(y)

∣∣∣2 ∫ |t|−n+2(1−s) dtdy

≤ C (s)(∫|Fu(z)|2

∣∣∣zβ

∣∣∣2 |z|2s dz+K∫ ∣∣∣Dβ u(y)

∣∣∣2 dy)

≤ C (s)∫ (

1+ |y|2)m+s

|Fu(y)|2 dy.

Since there are only finitely many such terms, this proves the lemma.