236 CHAPTER 9. BASIC FUNCTION SPACES

9. ↑ Now suppose f ∈ Lp(0,∞), p > 1, and f not necessarily in Cc(0,∞). Show thatF(x) = 1

x∫ x

0 f (t)dt still makes sense for each x > 0. Show the inequality of Problem8 is still valid. This inequality is called Hardy’s inequality. Hint: To show this, usethe above inequality along with the density of Cc (0,∞) in Lp (0,∞).

10. Suppose f ,g≥ 0. When does equality hold in Holder’s inequality?

11. Let α ∈ (0,1]. We define, for X a compact subset of Rp,

Cα (X ;Rn)≡ {f ∈C (X ;Rn) : ρα (f)+∥f∥ ≡ ∥f∥α< ∞}

where ∥f∥ ≡ sup{|f(x)| : x ∈ X} and

ρα (f)≡ sup{ |f(x)− f(y)||x−y|α

: x,y ∈ X , x ̸= y}.

Show that (Cα (X ;Rn) ,∥·∥α) is a complete normed linear space. This is called a

Holder space. What would this space consist of if α > 1?

12. Let {fn}∞n=1 ⊆Cα (X ;Rn) where X is a compact subset ofRp and suppose ∥fn∥α

≤Mfor all n. Show there exists a subsequence, nk, such that fnk converges in C (X ;Rn).We say the given sequence is precompact when this happens. (This also shows theembedding of Cα (X ;Rn) into C (X ;Rn) is a compact embedding.) Hint: You mightwant to use the Ascoli Arzela theorem, Theorem 9.2.4.

13. Let f : R×Rn→ Rn be continuous and bounded and let x0 ∈ Rn. If x : [0,T ]→ Rn

and h > 0, let

τhx(s)≡{

x0 if s≤ h,x(s−h) , if s > h.

For t ∈ [0,T ], let xh (t) = x0+∫ t

0 f(s,τhxh (s))ds. Show using the Ascoli Arzela theo-rem that there exists a sequence h→ 0 such that xh→ x in C ([0,T ] ;Rn). Next arguex(t) = x0 +

∫ t0 f(s,x(s))ds and conclude the following theorem. If f : R×Rn→ Rn

is continuous and bounded, and if x0 ∈ Rn is given, there exists a solution to thefollowing initial value problem.

x′ = f(t,x) , t ∈ [0,T ] , x(0) = x0.

This is the Peano existence theorem for ordinary differential equations.

14. Suppose f ∈ L∞∩L1. Show limp→∞ ∥ f∥Lp = ∥ f∥∞. Hint:

(∥ f∥∞− ε)p

µ ([| f |> ∥ f∥∞− ε])≤

∫[| f |>∥ f∥∞−ε]

| f |p dµ ≤

∫| f |p dµ =

∫| f |p−1 | f |dµ ≤ ∥ f∥p−1

∞

∫| f |dµ.

Now raise both ends to the 1/p power and take liminf and limsup as p→ ∞. Youshould get ∥ f∥

∞− ε ≤ liminf∥ f∥p ≤ limsup∥ f∥p ≤ ∥ f∥

∞

15. Suppose µ(Ω)<∞. Show that if 1≤ p< q, then Lq(Ω)⊆ Lp(Ω). Hint Use Holder’sinequality.