log-sobolev inequalities for infinite measures. I was wondering if we could have log-sobolev inequalities for infinite measures, most notably Lebesgue measure. I presume this is false, but I haven't been able to construct one. 
I tried playing with $1/x^{\alpha}$, and throwing some $log(x)$ and higher powers of it, but I haven't had much luck. 
Could someone provide me with a way to furnish such a counterexample? Or perhaps push me in the right direction?
Thanks!
EDIT: The inequality is as follows: For $f$ smooth, $f \in L^2$ and $\nabla f \in L^2$ , $\mu$ Gaussian measure on $\mathbb{R}^n$ we have:
$\int_\mathbb{R^n} f^2 \log f^2 d \mu \leq 2 \int_{\mathbb{R^n}} | \nabla f|^2  + ||f||^2_2 \log ||f||^2_2 $ 
 A: Actually the log-Sobolev inequality for Lebesgue measure $\mathscr{L}^n$ is true, up to losing a multiplicative constant.
In order to show it we recall that the original log-Sobolev inequality holds for the standard Gaussian measure $\mu=\frac 1{(2 \pi)^{n/2}} e^{-|x|^2/2}$ in the form (slightly different than what stated in the problem):
$$\int_{\mathbb{R}^n} f^2 \log f^2 d \mu - \left(\int_{\mathbb{R}^n} f^2 d\mu\right) \log \left(\int_{\mathbb{R}^n} f^2 d\mu\right) \leq 2 \int_{\mathbb{R}^n} | \nabla f|^2 d \mu.$$
Any reference is good: the original article of Gross http://www.jstor.org/stable/2373688?seq=1#page_scan_tab_contents as well as any survey about the argument (for example http://www.math.duke.edu/~rtd/CPSS2007/Berlin.pdf )
We will refer to this inequality as $LSI_{\mu}$. Now it is clear that this inequality is translation invariant and so, letting $\mu_y= \frac 1{(2 \pi)^{n/2}} e^{-|x-y|^2/2}$, also $LSI_{\mu_y}$ holds true.
Now let us consider a function $f \in H^1(\mathbb{R}^n, \mathscr{L}^n)$ and le $c^2=\int f^2 d \mathscr{L}^n$. Then we will have that, for every $y \in \mathbb{R}^d$, 
$$ \int_{\mathbb{R}^n} \Bigl( \frac{f}{c} \Bigr) ^2 d \mu_y \leq  \int_{\mathbb{R}^n} \Bigl( \frac{f}{c} \Bigr) ^2 d \mathscr{L}^n  =1; $$
in particular we have
$$\int_{\mathbb{R}^n}\Bigl( \frac{f}{c} \Bigr)^2 \log \Bigl( \frac{f}{c} \Bigr)^2 d \mu_y  \leq 2 \int_{\mathbb{R}^n} \left| \nabla \Bigl( \frac{f}{c} \Bigr)\right|^2 d \mu_y \qquad \forall y \in \mathbb{R}^d.$$
integrating this inequality for in $dy$ we obtain
$$\int_{\mathbb{R}^n}\Bigl( \frac{f}{c} \Bigr)^2 \log \Bigl( \frac{f}{c} \Bigr)^2 d \mathscr{L}^n  \leq 2 \int_{\mathbb{R}^n} \left| \nabla \Bigl( \frac{f}{c} \Bigr)\right|^2 d \mathscr{L}^n,$$
Now, multiplying the equation by $c^2$ and then exploiting the definition of $c$, we finally get
$$\int_{\mathbb{R}^n} f^2 \log f^2 d \mathscr{L}^n - \left(\int_{\mathbb{R}^n} f^2 d\mathscr{L}^n\right) \log \left(\int_{\mathbb{R}^n} f^2 d\mathscr{L}^n\right) \leq {2} \int_{\mathbb{R}^n} | \nabla f|^2 d \mathscr{L}^n.$$
