Ir al contenido principal

## Due March 23

### Problem 1

If $\{K_\delta\}$ is a family of better kernels, there exists a constant $A>0$ such that

$\sup_{\delta>0} |f*K_\delta(x)| \le A f^*(x)$

### Problem 2

For $a,b>0$, let

$\begin{cases} x^a\sin x^{-b} & 0 < x \le 1, \\ 0 & x=0.\end{cases}$

1. f is of bounded variation iff $a>b$.

2. For each $0<\alpha<1$, construct an $\alpha$-Hölder continuos function that is not of bounded variation.

3. If $a=b=2$, f' exists at every point but is not integrable.

### Problem 3

Define the one-sided maximal function for locally integrable functions on $\R$ as

$\displaystyle f_+^*(x) = \sup_{h>0} \frac{1}{h} \int_x^{x+h} |f|.$

If $E_\alpha^+ = \{x\in\R: f_+^*(x)>\alpha\}$, then

$\displaystyle m(E_\alpha^+) = \frac{1}{\alpha} \int_{E_\alpha^+} |f|.$

### Problem 4

Let $f:\R\to\R$ be absolutely continuous.

1. f maps sets of measure zero to sets of measure 0.

2. f maps measurable sets to measurable sets.