1308 CHAPTER 38. SOBOLEV SPACES BASED ON L2

Theorem 38.3.1 Let s ∈ (0,1) and let m be a nonnegative integer. Then an equivalentnorm for Hm+s (Rn) is

|||u|||2m+s ≡ ||u||2m,2,Rn + ∑

|α|=m

∫ ∫|Dα u(x)−Dα u(y)|2 |x−y|−n−2s dxdy.

Also if |β | ≤ m, there are constants, m(s) and M (s) such that

m(s)∫|Fu(z)|2

∣∣∣zβ

∣∣∣2 |z|2s dz≤∫ ∫ ∣∣∣Dβ u(x)−Dβ u(y)

∣∣∣2 |x−y|−n−2s dxdy

≤M (s)∫|Fu(z)|2

∣∣∣zβ

∣∣∣2 |z|2s dz (38.3.10)

Proof: Let u ∈S which is dense in Hm+s (Rn). The Fourier transform of the function,y→ Dα u(x+y)−Dα u(y) equals(

eix·z−1)

FDα u(z) .

Now by Fubini’s theorem and Plancherel’s theorem along with the above, taking |α|= m,∫ ∫|Dα u(x)−Dα u(y)|2 |x−y|−n−2s dxdy

=∫ ∫

|Dα u(y+ t)−Dα u(y)|2 |t|−n−2s dtdy

=∫|t|−n−2s

∫|Dα u(y+ t)−Dα u(y)|2 dydt

=∫|t|−n−2s

∫ ∣∣(eit·z−1)

FDα u(z)∣∣2 dzdt

=∫|FDα u(z)|2

(∫|t|−n−2s ∣∣(eit·z−1

)∣∣2 dt)

dz. (38.3.11)

Consider the inside integral, the one taken with respect to t.

G(z)≡(∫|t|−n−2s ∣∣(eit·z−1

)∣∣2 dt).

The essential thing to notice about this function of z is that it is a positive real numberwhenever z ̸= 0. This is because for small |t| , the integrand is dominated by C |t|−n+2(1−s) .Changing to polar coordinates, you see that∫

[|t|≤1]|t|−n−2s ∣∣(eit·z−1

)∣∣2 dt < ∞

Next, for |t|> 1, the integrand is bounded by 4 |t|−n−2s , and changing to polar coordinatesshows ∫

[|t|>1]|t|−n−2s ∣∣(eit·z−1

)∣∣2 dt ≤ 4∫[|t|>1]

|t|−n−2s dt < ∞.