416 CHAPTER 15. THE Lp SPACES

Show that E−E contains an interval. Hint: Let

f (x) =∫

XE(t)XE(x+ t)dt.

Note f is continuous at 0 and f (0)> 0 and use continuity of translation in Lp.

2. Establish the inequality || f g||r ≤ || f ||p ||g||q whenever 1r = 1

p +1q .

3. Let (Ω,S ,µ) be counting measure onN. Thus Ω =N and S =P (N) with µ (S) =number of things in S. Let 1≤ p≤ q. Show that in this case,

L1 (N)⊆ Lp (N)⊆ Lq (N) .

Hint: This is real easy if you consider what∫

Ωf dµ equals. How are the norms

related?

4. Consider the function, f (x,y) = xp−1

py + yq−1

qx for x,y> 0 and 1p +

1q = 1. Show directly

that f (x,y)≥ 1 for all such x,y and show this implies xy≤ xp

p + yq

q .

5. Give an example of a sequence of functions in Lp (R) which converges to zero in Lp

but does not converge pointwise to 0. Does this contradict the proof of the theoremthat Lp is complete?

6. Let K be a bounded subset of Lp (Rn) and suppose that there exists G such that G iscompact with ∫

Rn\G|u(x)|p dx < ε

p

and for all ε > 0, there exist a δ > 0 and such that if |h|< δ , then∫|u(x+h)−u(x)|p dx < ε

p

for all u ∈ K. Show that K is precompact in Lp (Rn). Hint: Let φ k be a mollifier andconsider

Kk ≡ {u∗φ k : u ∈ K} .

Verify the conditions of the Ascoli Arzela theorem for these functions defined on Gand show there is an ε net for each ε > 0. Can you modify this to let an arbitraryopen set take the place of Rn?

7. Let (Ω,d) be a metric space and suppose also that (Ω,S ,µ) is a regular measurespace such that µ (Ω) < ∞ and let f ∈ L1 (Ω) where f has complex values. Showthat for every ε > 0, there exists an open set of measure less than ε, denoted here byV and a continuous function, g defined on Ω such that f = g on VC. Thus, aside froma set of small measure, f is continuous. If | f (ω)| ≤M, show that it can be assumedthat |g(ω)| ≤ M. This is called Lusin’s theorem. Hint: Use Theorems 15.2.4 and15.1.10 to obtain a sequence of functions in Cc (Ω) ,{gn} which converges pointwisea.e. to f and then use Egoroff’s theorem to obtain a small set, W of measure less