38.3. AN INTRINSIC NORM 1309

Now for α > 0,

G(αz) =∫|t|−n−2s ∣∣(eit·αz−1

)∣∣2 dt

=∫|t|−n−2s ∣∣(eiαt·z−1

)∣∣2 dt

=∫ ∣∣∣ r

α

∣∣∣−n−2s ∣∣(eir·z−1)∣∣2 1

αn dr

= α2s∫|r|−n−2s ∣∣(eir·z−1

)∣∣2 dr = α2sG(z) .

Also G is continuous and strictly positive. Letting

0 < m(s) = min{G(w) : |w|= 1}

andM (s) = max{G(w) : |w|= 1} ,

it follows from this, and letting α = |z| ,w≡ z/ |z| , that

G(z) ∈(

m(s) |z|2s ,M (s) |z|2s).

More can be said but this will suffice. Also observe that for s ∈ (0,1) and b > 0,

(1+b)s ≤ 1+bs, 21−s (1+b)s ≥ 1+bs.

In what follows, C (s) will denote a constant which depends on the indicated quantitieswhich may be different on different lines of the argument. Then 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 top half of 38.3.10. Therefore,

|||u|||2m+s ≡ ||u||2m,2,Rn + ∑|α|=m

∫ ∫|Dα u(x)−Dα u(y)|2 |x−y|−n−2s dxdy

≤ C∫ (

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

∫|Fu(z)|2 ∑

|α|=m|zα |2 |z|2s dz

Recall that

∑|α|≤m

z2α11 · · ·z2αn

n ≤

(1+

n

∑j=1

z2j

)m

≤C (n,m) ∑|α|≤m

z2α11 · · ·z2αn

n . (38.3.12)