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Homework 7, Real Analysis 2

Due March 23

Problem 1

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

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

for all $latex f\in L^1$.

Problem 2

For $latex a,b>0$, let

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

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

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

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

Problem 3

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

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

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

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

Problem 4

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

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

  2. f maps measurable sets to measurable sets.