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Is log(log(1+1/r)) in L^N(B(0,1))?

  1. Dec 30, 2006 #1
    1. The problem statement, all variables and given/known data
    Ok, so we're in R^N, and we're looking at the unit ball. I wanna prove that the function g(x)=log(log(1+1/r)) is in L^N of the unit ball, where r=|x|.
    I also want to prove that its partial derivative are there, but like we say here, one cow at a time.:smile:


    2. Relevant equations
    None, really.


    3. The attempt at a solution
    Well, I don't remember if it's true, but maybe if xf(x) tends to zero when x tends to zero and f>0, then its integral is finite? I could then try and use it on |g|^N...
    Any help would be immensly appreciated.
     
  2. jcsd
  3. Dec 30, 2006 #2

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    What does L^N of the unit ball in R^N mean? If it means Lp(B^N), where B^N is the unit ball in R^N, and Lp(X) is the space of complex valued functions on X whose magnitude to the pth power is integrable, p>=1, then:

    Two steps:
    1. show (log(x))^p<x for all p>=1, all x>N, some N depending on p.
    2. show log(1+1/r) is integrable
     
    Last edited: Dec 30, 2006
  4. Dec 30, 2006 #3
    Thanks, success!:smile:
     
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