# Homework Help: Finiteness of an integral given an $L^2$ function

1. Mar 9, 2013

### lmedin02

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

2. Relevant equations

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