12.7. EXERCISES 373

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

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

√x and 1/x.

21. 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∥θ

r ∥ f∥1−θs .

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.

22. 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?

23. Establish the following for f ̸= 0 in Lp

∥ f∥p =∫

Ω

| f |p

∥ f∥p−1p

dµ =∫

Ω

f| f |p−2 f̄

∥ f∥p−1p

≤ sup∥g∥q≤1

∫| f | |g|dµ ≤ ∥ f∥p

24. ↑From the above problem, if f is nonnegative and product measurable,(∫ (∫f (x,y)dµ (x)

)p

dν (y))1/p

= sup∥h∥q≤1

∫ (∫f (x,y)dµ (x)

)h(y)dν (y)

Now use Fubini’s theorem and then the Holder inequality to obtain

= sup∥h∥q≤1

∫ ∫f (x,y)h(y)dν (y)dµ (x)≤

∫ (∫f (x,y)p dν (y)

)1/p

dµ (x)

This gives another proof of the important Minkowski inequality for integrals.

25. Let 0 < p < 1 and let f ,g be measurable C valued functions. Also∫Ω

|g|p/(p−1) dµ < ∞,∫

Ω

| f |p dµ < ∞

Then show the following backwards Holder inequality holds.

∫Ω

| f g|dµ ≥(∫

Ω

| f |p dµ

)1/p(∫Ω

|g|p/(p−1) dµ

)(p−1)/p

Hint: You should first note that g = 0 only on a set of measure zero. Then youcould write

∫Ω| f |p dµ =

∫Ω|g|−p | f g|p dµ. Apply the usual Holder inequality with

1/p one of the exponents and 1/(1− p) the other exponent. Then the above is ≤(∫Ω|g|−p/(1−p) dµ

)1−p(∫

Ω| f g|dµ)p etc. Note the usual Holder inequality in case

p > 1 is∫

Ω| f g|dµ ≤ (

∫Ω| f |p dµ)1/p

(∫Ω|g|p/(p−1) dµ

)(p−1)/p.