11.7. EXERCISES 291

11. Let f ∈ G and let k be a positive integer.

∥ f∥k,2 ≡ (∥ f∥22 + ∑|α|≤k∥Dα f∥2

2)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 G 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? Since they are equivalent norms, we usually just use ∥∥k,2or ∥∥Hk(Rn).

12. ↑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 se-quence in ∥ ∥k,2 of Problem 11}. This is one way to define Sobolev Spaces. Hint:One way to 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 Borel measure which is inner and outer regu-

lar and show there exists {gm} such that gm ∈ G and gm → F f in L2(µ). Thusgm = F fm, fm ∈ G because F maps G onto G . Then by Problem 11, { fm } is Cauchyin the norm ∥ ∥k,2. By using the countable version of G in which the polynomials allhave rational coefficients and in e−a|x|2 the a is a positive rational, show that Hk (Rn)is separable.

13. ↑ If 2k > n, show that if f ∈ Hk(Rn), then f equals a bounded continuous func-tion a.e. Hint: Show that for k this large, F f ∈ L1(Rn), and then use Problem 1 orTheorem 11.6.11. To do this, 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.

14. For u ∈ G , define γu(x′) ≡ u(x′,0). Show that there is a constant C independent ofu such that ∫

Rn−1

∣∣γu(x′)∣∣2 dx′ ≤C2 ∥u∥2

1,2

where this is the Sobolev norm described in Problem 11. Explain how this impliesthat one can give a meaningful description of the value of u on an n−1 dimensional