38.3. AN INTRINSIC NORM 1311

is continuous, linear, one to one, and has an inverse with the same properties, the inversebeing

(h−1)∗.

Proof: Let u∈S. From Theorem 38.2.7 and the equivalence of the norms in W m,2 (Rn)and Hm (Rn) ,

||h∗u||2Hm(Rn)+∫ ∫

∑|α|=m |Dα h∗u(x)−Dα h∗u(y)|2 |x−y|−n−2s dxdy

≤C ||u||2Hm(Rn)+∫ ∫

∑|α|=m |Dα h∗u(x)−Dα h∗u(y)|2 |x−y|−n−2s dxdy

=C ||u||2Hm(Rn)+∫ ∫

∑|α|=m

∣∣∣∑|β (α)|≤m h∗(

Dβ (α)u)

gβ (α) (x)

−h∗(

Dβ (α)u)

gβ (α) (y)∣∣∣2 |x−y|−n−2s dxdy

≤C ||u||2Hm(Rn)+C∫ ∫

∑|α|=m ∑|β (α)|≤m

∣∣∣h∗(Dβ (α)u)

gβ (α) (x)

−h∗(

Dβ (α)u)

gβ (α) (y)∣∣∣2 |x−y|−n−2s dxdy

(38.3.13)

A single term in the last sum corresponding to a given α is then of the form,∫ ∫ ∣∣∣h∗(Dβ u)

gβ (x)−h∗(

Dβ u)

gβ (y)∣∣∣2 |x−y|−n−2s dxdy (38.3.14)

≤[∫ ∫ ∣∣∣h∗(Dβ u

)(x)gβ (x)−h∗

(Dβ u

)(y)gβ (x)

∣∣∣2 |x−y|−n−2s dxdy +

∫ ∫ ∣∣∣h∗(Dβ u)(y)gβ (x)−h∗

(Dβ u

)(y)gβ (y)

∣∣∣2 |x−y|−n−2s dxdy]

≤[C (h)

∫ ∫ ∣∣∣h∗(Dβ u)(x)−h∗

(Dβ u

)(y)∣∣∣2 |x−y|−n−2s dxdy +

∫ ∫ ∣∣∣h∗(Dβ u)(y)∣∣∣2 ∣∣gβ (x)−gβ (y)

∣∣2 |x−y|−n−2s dxdy].

Changing variables, and then using the names of the old variables to simplify the notation,

≤[C(h,h−1)∫ ∫ ∣∣∣(Dβ u

)(x)−

(Dβ u

)(y)∣∣∣2 |x−y|−n−2s dxdy +

∫ ∫ ∣∣∣h∗(Dβ u)(y)∣∣∣2 ∣∣gβ (x)−gβ (y)

∣∣2 |x−y|−n−2s dxdy].

By 38.3.10,

≤ C (h)∫|F (u)(z)|2

∣∣∣zβ

∣∣∣2 |z|2s dz

+∫ ∫ ∣∣∣h∗(Dβ u

)(y)∣∣∣2 ∣∣gβ (x)−gβ (y)

∣∣2 |x−y|−n−2s dxdy.