Chapter 38

Sobolev Spaces Based On L2

38.1 Fourier Transform TechniquesMuch insight can be obtained easily through the use of Fourier transform methods. Thistechnique will be developed in this chapter. When this is done, it is necessary to useSobolev spaces of the form W k,2 (U) , those Sobolev spaces which are based on L2 (U) .It is true there are generalizations which use Fourier transform methods in the context ofLp but the spaces so considered are called Bessel potential spaces. They are not reallySobolev spaces. Furthermore, it is Mihlin’s theorem rather than the Plancherel theoremwhich 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 functionswhich 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 usedearlier in defining the Fourier transforms which are a suspace of S. Recall the followingdefinition.

Definition 38.1.1 f ∈S, the Schwartz class, if f ∈C∞(Rn) and for all positive integers N,

ρN( f )< ∞

whereρN( f ) = sup{(1+ |x|2)N |Dα f (x)| : x ∈ Rn , |α| ≤ N}.

Thus f ∈S if and only if f ∈C∞(Rn) and

sup{|xβ Dα f (x)| : x ∈ Rn}< ∞ (38.1.1)

for all multi indices α and β .

Thus all partial derivatives of a function in S are in Lp (Rn) for all p ≥ 1. Therefore,for f ∈S, the Fourier and inverse Fourier transforms are given in the usual way,

F f (t) =(

12π

)n/2 ∫Rn

f (x)e−it·xdx, F−1 f (t) =(

12π

)n/2 ∫Rn

f (x)eit·xdx.

Also recall that the Fourier transform and its inverse are one to one and onto maps from Sto S.

To tie the Fourier transform technique in with what has been done so far, it is necessaryto make the following assumption on the set, U. This assumption is made so that it ispossible to consider elements of W k,2 (U) as restrictions of elements of W k,2 (Rn) .

Assumption 38.1.2 Assume U satisfies the segment condition and that for any m of inter-est, there exists E ∈L (W m,p (U) ,W m,p (Rn)) such that for each

k ≤ m, E ∈L(

W k,p (U) ,W k,p (Rn)).

That is, there exists a stong (m, p) extension operator.

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