Download Math 525 Homework: Problems on Measurable Functions and Lebesgue Integrals and more Assignments Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Math 525
Homework due 01/31/
Reading:
Sections 1, 2, and 3 Chapter 8 from Royden. Sections 1 and 2, Chapter 4 from Folland.
Problems from Folland:
Chapter 3, Section 4: problems 25 and 26.
Problem 1. Show that there exists a constant C > 0 such that for every f ∈ L^1 loc(Rn)
m ({x ∈ Rn^ : M f (x) > α}) ≤
C
α
{x: |f (x)|≥ α 2 }
|f (y)| dy.
Recall that for x ∈ Rn
M f (x) = sup r> 0
m(B(x, r))
B(x,r)
|f (y)| d(y).
Problem 2. Let p ≥ 1, and let f : Rn^ → [0, ∞) be measurable. Let
At = {x ∈ Rn^ : f (x) > t}.
Show that ∫
A 0 ×(0,∞)
tp−^1 χAt (x) dx dt =
0
tp−^1 m(At) dt =
p
A 0
f p(x) dx
Problem 3. For p > 1, define
Lp(Rn) =
f : Rn^ → R :
Rn
|f (y)|p^ dy < ∞
Show that there exists a constant C(p) > 1 depending on p such that
(∫
Rn
[M f (x)]p^ dx
) (^1) p ≤ C(p)
Rn
|f |(x)p^ dx
) (^1) p ,
for all f ∈ Lp(Rn). (Hint: Use problems 1 and 2.)
Problem 4. Let Ln^ denote Lebesgue measure on Rn^ and let μ be a Borel regular measure on Rn. Assume that for all x ∈ Rn
lim sup r→ 0
μ(B(x, r)) ωnrn^
where ωn denotes the Lebesgue measure of the ball of center 0 and radius 1 in Rn. Prove that for any Borel subset A ⊂ Rn μ(A) ≥ Ln(A).
∗Problem. A positive Radon measure μ on Rn^ is said to be a doubling measure if there exist C 0 > 1 such that
μ(B(x, 2 r)) ≤ C 0 μ(B(x, r)), ∀x ∈ Rn^ and ∀r > 0.
For f ∈ L^1 loc(μ) define the Hardy-Littlewood maximal function M f by
M f (x) = sup r> 0
μ(B(x, r))
B(x,r)
|f (y)| dμ(y).
Let μ be a doubling measure on Rn. Show that there exists a constant C > 0 such that for all f ∈ L^1 (μ) and all α > 0,
μ ({x ∈ Rn^ : M f (x) > α}) ≤
C
α
Rn
|f (y)| dμ(y).
