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Finiteness of an integral given an [itex]L^2[/itex] function

  1. Mar 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Let [itex]\Omega[/itex] be a torus and [itex]g\in L^2(\Omega)[/itex] be a scalar value function. Is [itex]\int_{\Omega}{e^g}dx<\infty[/itex]?

    2. Relevant equations

    3. The attempt at a solution Not sure where to start. However, if [itex]g\in W^{1,2}(\Omega)[/itex] then I can show that the answer is yes by applying Moser-Trudinger inequality and decomposing [itex]W^{1,2}[/itex] as the direct sum of [itex]\mathbb{R}[/itex] and functions of average value zero on [itex]W^{1,2}[/itex]. Any hints are greatly appreciated.
  2. jcsd
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