9.7. EXERCISES 237

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

√x and 1/x.

17. 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 = θq

r + q(1−θ)s and then use

Holder’s inequality.

18. 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?

19. Let T be a real number, T < 1. Let A0 = 0, An+1 = An +12

(T −A2

n). Show that

An ∈[0, 1+T

2

]. Use the mean value theorem to show that f (x)≡ x+ 1

2

(T − x2

)maps[

0, 1+T2

]to[0, 1+T

2

]and is a contraction map. Obtain a unique square root for T as

a fixed point.