32.4. EXERCISES 1117

which equals

2eγt

π

∫∞

0

g(t−u)e−γ(t−u)+g(t +u)e−γ(t+u)

2sin(Ru)

udu

and then apply the result of Problem 6.

9. Suppose f ∈S. Show F( fx j)(t) = it jF f (t).

10. Let f ∈S and let k be a positive integer.

|| f ||k,2 ≡ (|| f ||22 + ∑|α|≤k||Dα f ||22)1/2.

One could also define

||| f |||k,2 ≡ (∫

Rn|F f (x)|2(1+ |x|2)kdx)1/2.

Show both || ||k,2 and ||| |||k,2 are norms on S and that they are equivalent. Theseare Sobolev space norms. For which values of k does the second norm make sense?How about the first norm?

11. ↑ Define Hk(Rn),k ≥ 0 by f ∈ L2(Rn) such that

(∫|F f (x)|2(1+ |x|2)kdx)

12 < ∞,

||| f |||k,2 ≡ (∫|F f (x)|2(1+ |x|2)kdx)

12.

Show Hk(Rn) is a Banach space, and that if k is a positive integer, Hk(Rn) ={ f ∈L2(Rn) : there exists {u j} ⊆ G with ||u j− f ||2→ 0 and {u j} is a Cauchy sequencein || ||k,2 of Problem 10}. This is one way to define Sobolev Spaces. Hint: One wayto do the second part of this is to define a new measure, µ by

µ (E)≡∫

E

(1+ |x|2

)kdx.

Then show µ is a Radon measure and show there exists {gm} such that gm ∈ G andgm → F f in L2(µ). Thus gm = F fm, fm ∈ G because F maps G onto G . Then byProblem 10, { fm } is Cauchy in the norm || ||k,2.

12. ↑ If 2k > n, show that if f ∈ Hk(Rn), then f equals a bounded continuous functiona.e. Hint: Show that for k this large, F f ∈ L1(Rn), and then use Problem 1. To dothis, write

|F f (x)|= |F f (x)|(1+ |x|2)k2 (1+ |x|2)

−k2 ,

So ∫|F f (x)|dx =

∫|F f (x)|(1+ |x|2)

k2 (1+ |x|2)

−k2 dx.

Use the Cauchy Schwarz inequality. This is an example of a Sobolev imbeddingTheorem.