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## 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 $\delta\to 0$.