Real Analysis: Hardy Littlewood

[nateHI]

Homework Statement

Establish the Inequality $f^*(x)\ge \frac{c}{|x|ln\frac{1}{x}}$ for
$f(x)=\frac{1}{|x|(ln\frac{1}{x})^2}$ if $|x|\le 1/2$ and 0 otherwise

Homework Equations

$f^*(x)=\sup_{x\in B} \frac{1}{m(B)} \int_B|f(y)|dy \quad x\in \mathbb{R}^d$

The Attempt at a Solution

Disregard, I figured it out.

 

[jedishrfu]

What was the sticking point that you overcame?

 

[nateHI]

I was stuck on the first step. I was able to work in reverse from the solution but felt like I was missing a key idea doing it that way. Namely,

$\sup_{x\in B} \frac{1}{m(B)} \int_B \frac{1}{|x|(ln\frac{1}{x})^2}\ge \frac{1}{2|x|}\int_{-|x|}^{|x|} \frac{1}{|x|(ln\frac{1}{x})^2}$

from there you just work out the integral.