Homework 3

October 25, 2023

Homework set 3 due Friday, Oct. 31

The answers to the first three problems below should be quite brief.

Problem 1

Briefly answer the three questions (why?) in the following argument. Please label your answers A,B,C. Even though B seems obvious, give a convincing reason for it too.1

Let ν be a complex measure on (X, M). Suppose μ is any finite positive measure such that dν = fdμ for some f ∈ L1(μ)2 Such a μ always exists, for example, μ := |νr| + |νi|. We define the total variation measure of ν by

d|ν| = |f|dμ

One must check this is well-defined. Suppose dν = f1dμ1 = f2dμ2. Let λ = μ1 + μ2, so

there exist gi ≥ 0 such that

Thus,

so

dμi = gidλ,i = 1,2. (A. why?)

f1g1dλ = dν = f2g2dλ (B. why?),

|f1| dμ1 = |f1| g1dλ = |f2| g2dλ = |f2| dμ2. (C. why?) Thus, |ν| is well-defined.

1”Substitution” is not convincing for B.

2Weoftenwritedν=fdμasν=fdμ. Thismeansν(E)=R fdμforE∈M. Wesaythatfisthe

E

”Radon-Nikodym derivative of ν with respect to μ ”, and sometimes write f = dν/dμ.

1

Problem 2

In the case where ν is a finite signed measure, show that the definition of |ν| in Problem 1 agrees with our earlier definition: |ν| = ν+ + ν−. Hint: Make sense of, prove, and then use:

dν=(χP −χN)d|ν|

Problem 3

Let ν be a regular signed or complex Borel measure on Rn (”regular” here is defined on page 99 of the posted Folland text), and let dν = λ + f dm be its Lebesgue-Radon–Nikodym decomposition.

Show d|ν| = |λ| + |f|dm. Problem 4

Let χBr denote the characteristic function of Br = {x ∈ Rn : |x| ≤ r}. Set ψr(x) = 1 χBr (x)

pick g ∈ L1 (R+),g ≥ 0, and set

φ(x) = Define the maximal function

ψs(x)g(s)ds, φr(x) = r−nφ r

Show that

m(Br)

Z∞ ?x?

0

Mφf(x) = sup r>0

Z

Mφf(x) ≤ ∥g∥L1Mf(x)

where Mf(x) is the ”Hardy-Littlewood maximal function,” defined by

Problem 5

1Z

M(f)(x) = sup m (B ) |f(y)|dy.

r>0 r Br(x)

φr(x − y)|f(y)|dy

In the setting of Problem 4, show that

f ∈L1(Rn,dx)=⇒limφr ∗f(x)=Af(x), a.e.

where A = R ∞ g(s)ds 0

r→0

2