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(Solved) Real Analysis: Hardy Littlewood

  1. Dec 5, 2014 #1
    1. The problem statement, all variables and given/known data
    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

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

    3. The attempt at a solution
    Disregard, I figured it out.
     
    Last edited: Dec 5, 2014
  2. jcsd
  3. Dec 6, 2014 #2

    jedishrfu

    Staff: Mentor

    What was the sticking point that you overcame?
     
  4. Dec 7, 2014 #3
    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.
     
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