10.9. EXERCISES 263

5. Let φ m ∈C∞c (Rn),φ m (x)≥ 0, and

∫Rn φ m(y)dy = 1 with

limm→∞

sup{|x| : x ∈ spt(φ m)}= 0.

Show if f ∈ Lp(Rn), limm→∞ f ∗φ m = f in Lp(Rn). Hint: Use Minkowski’s inequal-ity for integrals to get a short proof of this fact.

6. Let 1p +

1p′ = 1, p > 1, let f ∈ Lp(Rn), g ∈ Lp′(Rn). Show f ∗g is uniformly contin-

uous on R and |( f ∗g)(x)| ≤ ∥ f∥Lp∥g∥Lp′ . f ∗g(x) ≡∫Rn f (x−y)g(y)dmn. Hint:

You need to consider why f ∗ g exists and then this follows from the definition ofconvolution and continuity of translation of Lebesgue measure.

7. Suppose f is a strictly decreasing nonnegative function defined on [0,∞).Let f−1 (y)≡ {x : such that y ∈ [ f (x+) , f (x−)]}. Show that∫

∞

0f (t)dt =

∫ f (0)

0f−1 (y)dy

Hint: Try to show that f−1 (y) = m([ f > y]) .

8. Let f (y) = g(y) = |y|−1/2 if y ∈ (−1,0)∪ (0,1) and f (y) = g(y) = 0 outside of thisset. For which values of x does it make sense to write the integral∫

Rf (x− y)g(y)dy≡ f ∗g(x)

This is asking for you to find where the convolution of f and g makes sense.

9. Let f ∈ L1 (Rp) and let g ∈ L1 (Rp) . Define the convolution of f and g as follows. Itequals

f ∗g(x)≡∫

f (x−y)g(y)dmp

Show that the above integral makes sense for a.e. x that is, for all x off a set ofmeasure zero. If f ∗ g is defined to equal 0 at points where the above integral doesnot make sense, show that ∥ f ∗g∥1 ≤ ∥ f∥1 ∥g∥1 where ∥h∥1 ≡

∫|h|dmp.

10. Consider D ≡ {p(e−αt)} where p(t) is some real polynomial having zero constantterm and α is some positive number at least as large as a given α0 > 0. Showthat D is an algebra and is dense in C0 ([0,∞)) with respect to the norm ∥ f∥

∞≡

max{| f (x)| : x ∈ [0,∞)}.

11. ↑Suppose f ∈ L1 ([0,∞)) and∫

∞

0 f (t)g(t)dm= 0 for all g∈D in the above problem.Explain why f (t) = 0 a.e. t. Hint: You can assume f is real since if not, you couldlook at the real and imaginary parts. You can also assume that f is nonnegative.Show density of Cc ([0,∞)) in L1 ([0,∞)). Then show there is a sequence of things inD which converges in L1 to f . Finally, go over why you can get a further subsequencewhich converges to f a.e. Then use Fatou’s lemma.

12. A measurable function f defined on [0,∞) has exponential growth if f (t) ≤ Cert

for some real r. Suppose you have f measurable with exponential growth. ShowL f (s)≡

∫∞

0 e−st f (t)dt the Laplace transform, is well defined for all s large enough.Now show that if L f (s)= 0 for all s large enough, then f (t)= 0 for a.e. t. This showsthat if two measurable functions with exponential growth have the same Laplacetransform for large s, then they are a.e. the same function.