Homework 6, Real Analysis 2

Due March 16

Problem 1

Consider the function on \R given by

f(x) = \begin{cases}\dfrac{1}{|x|(\log |x|)^2} & |x|\le 1/2\\0 & \text{otherwise.}\end{cases}

  1. f is integrable.
  2. f^*(x) \ge \dfrac{c}{|x|\log 1/|x|} for some c>0 and all |x|\le 1/2.
  3. f^* is not locally integrable.

Problem 2

Let E\subset [0,1] be a measurable set such that there exists \alpha > 0 such that

m(E\cap I) \ge \alpha m(I)

for all intervals I\subset[0,1]. Then m(E)=1.

Problem 3

Let F\subset\R be closed and \delta(x) the distance from x to F. Then

\delta(x+y) =o(|y|)

for almost every x\in F.

Problem 4

Suppose \{K_\delta\} is a family of kernels that satisfies

  • \displaystyle \int_\R K_\delta = 0 for all \delta>0.
  • For some A>0, |K_\delta(x)| \le A \min\{\delta^{-d}, \delta/|x|^{d+1}\} for all \delta > 0, x\in\R^d.

If f\in L^1(\R), then

f*K_\delta(x) \to 0

for a.e. x as \delta\to 0.

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