Math 526
Homework due 05/23/07
Reading: Folland: Chapter 8, sections 2 and 3.
Problems from Folland: Chapter 8, section 2: problem 8, 9.
Problem 1: Show by an example that C[0,1] is NOT dense in L∞[0,1]
Problem 2 [Weyl’s lemma]: Let u∈L1(Ω) where Ω is an open connected subset of Rn.
Assume that uis weakly harmonic in Ω, that is
ZΩ
u∆ψ= 0 ∀ψ∈C∞
c(Ω).
Show that (after redefinition on a set of measure 0) uis a C∞(Ω) harmonic function. (Hint:
use an approximate identity).
Problem 3 (Fall 1992): For f∈L2([0,∞)) define the function T f on [0,∞) by
T f (x) = Z∞
x
f(t)
t+ 1 dt.
1. Show that Tis a one-to-one bounded linear map from L2([0,∞)) to C0([0,∞)) (the
space of continuous functions on [0,∞) vanishing at infinity, with the supremum norm),
and that kTk= 1. (Hint: start by showing that the integral defining T f(x) always
converges.)
2. Show that the image of the unit ball in L2([0,∞)) under Thas compact closure in
C0([0,∞)).
Problem 4 (Fall 93): Suppose that f, g ∈L1(R). Show that
lim
y→∞ Z∞
−∞
|f(x) + g(x−y)|dx =Z∞
−∞
|f(x)|dx +Z∞
−∞
|g(x)|dx.
1