15.6. EXERCISES 419

21. 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 ||∞

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

23. Show L1(R)⊈ L2(R) and L2(R)⊈ L1(R) if Lebesgue measure is used. Hint: Con-sider 1/

√x and 1/x.

24. Suppose that θ ∈ [0,1] and r,s,q > 0 with

1q=

θ

r+

1−θ

s.

show that(∫| f |qdµ)1/q ≤ ((

∫| f |rdµ)1/r)θ ((

∫| f |sdµ)1/s)1−θ.

If q,r,s≥ 1 this says that|| f ||q ≤ || f ||θr || f ||1−θ

s .

Using this, show that

ln(|| f ||q

)≤ θ ln(|| f ||r)+(1−θ) ln(|| f ||s) .

Hint: ∫| f |qdµ =

∫| f |qθ | f |q(1−θ)dµ.

Now note that 1 = θqr + q(1−θ)

s and use Holder’s inequality.

25. Suppose f is a function in L1 (R) and f is infinitely differentiable. Is f ′ ∈ L1 (R)?Hint: What if φ ∈C∞

c (0,1) and f (x) = φ (2n (x−n)) for x ∈ (n,n+1) , f (x) = 0 ifx < 0?