1316 CHAPTER 38. SOBOLEV SPACES BASED ON L2
Lemma 38.5.3 Let u ∈S. Then there exists a constant, Cn, depending on n but indepen-dent of u ∈S such that
Fγu(x′)=Cn
∫R
Fu(x′,xn
)dxn.
Proof: Using the dominated convergence theorem,∫R
Fu(x′,xn
)dxn ≡ lim
ε→0
∫R
e−(εxn)2Fu(x′,xn
)dxn
≡ limε→0
∫R
e−(εxn)2(
12π
)n/2 ∫Rn
e−i(x′·y′+xnyn)u(y′,yn
)dy′dyndxn
= limε→0
(1
2π
)n/2 ∫Rn
u(y′,yn
)e−ix′·y′
∫R
e−(εxn)2e−ixnyndxndy′dyn.
Now −(εxn)2− ixnyn =−ε2
(xn +
iyn2
)2− ε2 y2
n4 and so the above reduces to
limε→0
(1
2π
)n/2 ∫Rn
u(y′,yn
)e−ix′·y′
∫R
e−ε2(
xn+iyn2
)2−ε2 y2
n4 dxndy′dyn
= limε→0
Kn
∫Rn
u(y′,yn
)e−ix′·y′e−ε2 y2
n4
∫R
e−ε2(
xn+iyn2
)2
dxndy′dyn
= limε→0
Kn
∫Rn
u(y′,yn
)e−ix′·y′e−ε2 y2
n4
1ε
dy′dyn
which is an expression of the form
limε→0
Kn
∫R
1ε
e−ε2 y2n4
∫Rn−1
u(y′,yn
)e−ix′·y′dy′dyn = Kn
∫Rn
u(y′,0
)e−ix′·y′dy′
= KnFγu(x′)
and this proves the lemma with Cn ≡ K−1n .
Earlier Ht (Rn) was defined and then for U an open subset of Rn, Ht (U) was defined tobe the space of restrictions of functions of Ht (Rn) to U and a norm was given which madeHt (U) into a Banach space. The next task is to considerRn−1×{0} , a smaller dimensionalsubspace of Rn and examine the functions defined on this set, denoted by Rn−1 for shortwhich are restrictions of functions in Ht (Rn) . You note this is somewhat different becauseheuristically, the dimension of the domain of the function is changing. An open set in Rn
is considered an n dimensional thing but Rn−1 is only n− 1 dimensional. I realize this isvague because the standard definition of dimension requires a vector space and an open setis not a vector space. However, think in terms of fatness. An open set is fat in n directionswhereas Rn−1 is only fat in n− 1 directions. Therefore, something interesting is likely tohappen.
Let S denote the Schwartz class of functions on Rn and S′ the Schwartz class offunctions on Rn−1. Also, y′ ∈ Rn−1 while y ∈ Rn. Let u ∈ S. Then from Lemma 38.5.3