1314 CHAPTER 38. SOBOLEV SPACES BASED ON L2

Corollary 38.3.5 Let t = m+ s for s ∈ [0,1) and let U,V be open sets. Let φ ∈Cm,1c (V ).

This means spt(φ) ⊆ V and φ ∈Cm,1 (Rn) . Then if u ∈ Ht (U) then uφ ∈ Ht (U ∩V ) and||uφ ||Ht (U∩V ) ≤Cφ ||u||Ht (U) .

Proof: Let v|U = u a.e. where v ∈ Ht (Rn) . Then by Lemma 38.3.4, φv ∈ Ht (Rn) andφv|U∩V = φu a.e. Therefore, φu ∈ Ht (U ∩V ) and

||φu||Ht (U∩V ) ≤ ||φv||Ht (Rn) ≤Cφ ||v||Ht (Rn) .

Taking the infimum for all such v whose restrictions equal u, this yields

||φu||Ht (U∩V ) ≤Cφ ||u||Ht (U) .

This proves the corollary.

38.4 Embedding TheoremsThe Fourier transform description of Sobolev spaces makes possible fairly easy proofs ofvarious embedding theorems.

Definition 38.4.1 Let Cmb (Rn) denote the functions which are m times continuously differ-

entiable and for which

sup|α|≤m

supx∈Rn|Dα u(x)| ≡ ||u||Cm

b (Rn) < ∞.

For U an open set, Cm(U)

denotes the functions which are restrictions of Cmb (Rn) to U.

It is clear this is a Banach space, the proof being a simple exercise in the use of thefundamental theorem of calculus along with standard results about uniform convergence.

Lemma 38.4.2 Let u∈S and let n2 +m < t. Then there exists C independent of u such that

||u||Cmb (Rn) ≤C ||u||Ht (Rn) .

Proof: Using the fact that the Fourier transform maps S to S and the definition of theFourier transform,

|Dα u(x)| ≤ C ||FDα u||L1(Rn)

= C∫|xα | |Fu(x)|dx

≤ C∫ (

1+ |x|2)|α|/2

|Fu(x)|dx

≤ C∫ (

1+ |x|2)m/2(

1+ |x|2)−t/2(

1+ |x|2)t/2|Fu(x)|dx

≤ C(∫ (

1+ |x|2)m−t

dx)1/2(∫ (

1+ |x|2)t|Fu(x)|t

)1/2

≤ C ||u||Ht (Rn)

because for the given values of t and m the first integral is finite. This follows from a useof polar coordinates. Taking sup over all x ∈ Rn and |α| ≤ m, this proves the lemma.