15.6. EXERCISES 417

than ε/2 such that convergence is uniform on WC. Now let F be a closed subset ofWC such that µ

(WC \F

)< ε/2. Let V = FC. Thus µ (V ) < ε and on F = VC, the

convergence of {gn} is uniform showing that the restriction of f to VC is continuous.Now use the Tietze extension theorem.

8. Let φ m ∈C∞c (Rn),φ m (x)≥ 0,and

∫Rn φ m(y)dy = 1 with

limm→∞

sup{|x| : x ∈ spt(φ m)}= 0.

Show if f ∈ Lp(Rn), limm→∞ f ∗φ m = f in Lp(Rn).

9. Let φ : R→ R be convex. This means

φ(λx+(1−λ )y)≤ λφ(x)+(1−λ )φ(y)

whenever λ ∈ [0,1]. Verify that if x < y < z, then φ(y)−φ(x)y−x ≤ φ(z)−φ(y)

z−y and thatφ(z)−φ(x)

z−x ≤ φ(z)−φ(y)z−y . Show if s ∈ R there exists λ such that φ(s) ≤ φ(t)+λ (s− t)

for all t. Show that if φ is convex, then φ is continuous.

10. ↑ Prove Jensen’s inequality. If φ : R→ R is convex, µ(Ω) = 1, and f : Ω→ R is inL1(Ω), then φ(

∫Ω

f du)≤∫

Ωφ( f )dµ . Hint: Let s =

∫Ω

f dµ and use Problem 9.

11. Let 1p +

1p′ = 1, p> 1, let f ∈ Lp(R), g∈ Lp′(R). Show f ∗g is uniformly continuous

on R and |( f ∗ g)(x)| ≤ || f ||Lp ||g||Lp′ . Hint: You need to consider why f ∗ g existsand then this follows from the definition of convolution and continuity of translationin Lp.

12. B(p,q) =∫ 1

0 xp−1(1− x)q−1dx,Γ(p) =∫

∞

0 e−tt p−1dt for p,q > 0. The first of theseis called the beta function, while the second is the gamma function. Show a.) Γ(p+1) = pΓ(p); b.) Γ(p)Γ(q) = B(p,q)Γ(p+q).

13. Let f ∈Cc(0,∞) and define F(x) = 1x∫ x

0 f (t)dt. Show

||F ||Lp(0,∞) ≤p

p−1|| f ||Lp(0,∞) whenever p > 1.

Hint: Argue there is no loss of generality in assuming f ≥ 0 and then assume this isso. Integrate

∫∞

0 |F(x)|pdx by parts as follows:

∫∞

0F pdx =

show = 0︷ ︸︸ ︷xF p|∞0 − p

∫∞

0xF p−1F ′dx.

Now show xF ′ = f −F and use this in the last integral. Complete the argument byusing Holder’s inequality and p−1 = p/q.

14. ↑ 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 Problem13 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,∞).