374 CHAPTER 12. THE Lp SPACES

26. Let 0 < p < 1 and suppose∫|h|p dµ < ∞ for h = f ,g. Then(∫

(| f |+ |g|)p dµ

)1/p

≥(∫| f |p dµ

)1/p

+

(∫|g|p dµ

)1/p

This is the backwards Minkowski inequality. Hint: First explain why, since p <1,(| f |+ |g|)p ≤ | f |p + |g|p. It follows from this that∫

Ω

((| f |+ |g|)p−1

)p/(p−1)dµ < ∞

since∫|h|p dµ < ∞ for h = f ,g. Then (| f |+ |g|)p−1 plays the role of |g| in the above

backwards Holder inequality. Next do this:∫Ω

(| f |+ |g|)p dµ =∫

Ω

(| f |+ |g|)p−1 (| f |+ |g|)dµ

=∫

Ω

(| f |+ |g|)p−1 | f |dµ +∫

Ω

(| f |+ |g|)p−1 |g|dµ

Now apply the backwards Holder inequality of the above problem. The first termgives ∫

Ω

(| f |+ |g|)p−1 | f |dµ ≥(∫

Ω

(| f |+ |g|)p dµ

)(p−1)/p(∫Ω

| f |p dµ

)1/p

27. Let f ∈ L1loc (R) . Show there exists a set of measure zero N such that if x /∈ N, then

if {In} is a sequence of intervals containing x such that m(In)→ 0 then

1m(In)

∫In| f − f (x)|dx→ 0.

Generalize to higher dimensions if possible. Also, does In have to be an interval?

28. Suppose F (x) =∫ x

a f (t)dt so that F is absolutely continuous where f ∈ L1 ([a,b]).Show that f ∈ Lp for p > 1 if and only if there is M < ∞ such that whenevera = x0 < x1 < · · ·< xn = b it follows that ∑

ni=1|F(xi)−F(xi−1)|p

(xi−xi−1)p−1 < M. This problem is

a result of F. Riesz. Hint: The first part is an easy application of Holder’s inequality.For the second, let Pn be a sequence of paritions of [a,b] such that the subinter-

vals have lengths converging to 0. Define fn (x)≡ ∑nk=1

F(xnk)−F(xn

k−1)xn

k−xnk−1

XInk(x) where

the intervals of Pn are Ink =

[xn

k−1,xnk+1

]. Then for a.e. x, fn (x)→ f (x) thanks

to the Lebesgue fundamental theorem of calculus and Problem 27. Now apply Fa-tou’s lemma to say that

∫ ba | f (x)|

p dx ≤ liminfn→∞

∫ ba | fn (x)|p dx and simplify this

last integral by breaking it into a sum of integrals over the sub-intervals of Pn.

Note |F(xnk)−F(xn

k−1)|p

(xnk−xn

k−1)p does not depend on x ∈ In

k .

29. If f ∈ Lp (U,mp) , where U is a bounded open set in Rn, show there exists a sequenceof smooth functions which converges to f a.e. Then show there exists a sequence ofpolynomials whose restriction to U converges a.e. to f on U .

30. In Corollary 12.4.3 can you generalize where f is only in L1loc (Rp,mp).