1306 CHAPTER 38. SOBOLEV SPACES BASED ON L2

Definition 38.2.1 For m an integer and s ∈ (0,1) , let Hm+s (Rn)≡{u ∈ L2 (Rn) : ||u||Hm+s(Rn) ≡

(∫Rn

(1+ |x|2

)m+s|Fu(x)|2 dx

)1/2

< ∞

}. (38.2.7)

You could also simply refer to Ht (Rn) where t is a real number replacing the m+ s inthe above formula with t but I want to emphasize the notion that t = m+ s where m is anonnegative integer. Therefore, I will often write m+ s. Let U be an open set in Rn. Then

Hm+s (U)≡{

u : u = v|U for some v ∈ Hm+s (Rn)}. (38.2.8)

Define the norm in this space by

||u||Hm+s(U) ≡ inf{||v||Hm+s(Rn) : v|U = u

}. (38.2.9)

Lemma 38.2.2 Hm+s (U) is a Banach space.

Proof: Just repeat the proof of Lemma 38.1.7.The theorem about density of S also remains true in Hm+s (Rn) . Just repeat the proof

of that part of Lemma 38.1.5 replacing the integer, m, with the symbol, m+ s.

Lemma 38.2.3 S is dense in Hm+s (Rn).

In fact, more can be said.

Corollary 38.2.4 Let U be an open set and let S|U denote the restrictions of functions ofS to U. Then S|U is dense in Ht (U) .

Proof: Let u ∈ Ht (U) and let v ∈ Ht (Rn) such that v|U = u a.e. Then since S is densein Ht (Rn) , there exists w ∈S such that

||w− v||Ht (Rn) < ε.

It follows that

||u−w||Ht (U) ≤ ||u− v||Ht (U)+ ||v−w||Ht (U)

≤ 0+ ||v−w||Ht (Rn) < ε.

These fractional order spaces are important when trying to understand the trace on theboundary. The Fourier transform description also makes it very easy to establish interestinginequalities such as interpolation inequalities.

Lemma 38.2.5 Let 0≤ r < s < t. Then if u ∈ Ht (Rn) ,

||u||Hs(Rn) ≤ ||u||θ

Hr(Rn) ||u||1−θ

Ht (Rn)

where θ is a positve number such that θr+(1−θ) t = s.