Homework 7, Real Analysis 2

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)

for all f\in L^1.

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=2f’ 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.
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