Much insight can be obtained easily through the use of Fourier transform methods. This
technique will be developed in this chapter. When this is done, it is necessary to use
Sobolev spaces of the form W^{k,2}
(U )
, those Sobolev spaces which are based on L^{2}
(U )
. It
is true there are generalizations which use Fourier transform methods in the
context of L^{p} but the spaces so considered are called Bessel potential spaces. They
are not really Sobolev spaces. Furthermore, it is Mihlin’s theorem rather than
the Plancherel theorem which is the main tool of the analysis. This is a hard
theorem.
It is convenient to consider the Schwartz class of functions,S. These are functions
which have infinitely many derivatives and vanish quickly together with their derivatives
as
|x |
→∞. In particular, C_{c}^{∞}
(ℝn)
is contained in S which is not true of the functions,
G used earlier in defining the Fourier transforms which are a suspace of S. Recall the
following definition.
Definition 38.1.1f ∈S, the Schwartz class, if f ∈ C^{∞}(ℝ^{n}) and for all positiveintegers N,
ρ (f) < ∞
N
where
2 N α n
ρN(f) = sup{(1+ |x|) |D f(x)| : x ∈ ℝ ,|α| ≤ N }.
Thus f ∈S if and only if f ∈ C^{∞}(ℝ^{n}) and
sup{|x βDαf(x)| : x ∈ ℝn} < ∞ (38.1.1)
(38.1.1)
for all multi indices α and β.
Thus all partial derivatives of a function in S are in L^{p}
n
(ℝ )
for all p ≥ 1. Therefore,
for f ∈S, the Fourier and inverse Fourier transforms are given in the usual
way,
( 1-)n ∕2∫ −it⋅x − 1 (-1-)n∕2∫ it⋅x
F f (t) = 2π ℝn f (x)e dx,F f (t) = 2π ℝn f (x)e dx.
Also recall that the Fourier transform and its inverse are one to one and onto maps from
S to S.
To tie the Fourier transform technique in with what has been done so far, it is
necessary to make the following assumption on the set, U. This assumption is made so
that it is possible to consider elements of W^{k,2}
(U )
as restrictions of elements of
W^{k,2}
(ℝn)
.
Assumption 38.1.2Assume U satisfies the segment condition and that forany m of interest, there exists E ∈ ℒ
(W m,p(U),W m,p(ℝn))
such that for eachk ≤ m,E ∈ℒ
(W k,p(U ),W k,p(ℝn))
. That is, there exists a stong
(m,p)
extensionoperator.
Lemma 38.1.3The Schwartz class, S, is dense in W^{m,p}
(ℝn )
.
Proof:The set, ℝ^{n} satisfies the segment condition and so C_{c}^{∞}
(ℝn)
is dense in
W^{m,p}
(ℝn)
. However, C_{c}^{∞}
(ℝn)
⊆S. This proves the lemma.
Recall now Plancherel’s theorem which states that
||f||
_{0,2,ℝn} =
||Ff||
_{0,2,ℝn} whenever
f ∈ L^{2}
(ℝn)
. Also it is routine to verify from the definition of the Fourier transform that
for u ∈S,
F∂ku = ixkF u.
From this it follows that
||Dαu ||0,2,ℝn = ||xαFu||0,2,ℝn .
Here x^{α} denotes the function x → x^{α}. Therefore,