For $f\in C^{1}(\mathbb{R}^{n})$ Gross's sobolev inequality says that

$$\int f^{2} log f^{2}d\mu -\int f^{2}d\mu log(\int f^{2}d\mu)\leq \frac{2}{c}\int |\triangledown f|^{2}d\mu,$$

where $d\mu=\frac{1}{\pi^{n/2}}e^{-|x|^{2}}dx$ for some $c>0$. Any suggestions? Can someone explain the variational argument proposed in the linked paper.

Attempt

Proof

We first prove it for n=1 and then finish by induction. We may assume $\int u^{2}d\mu=1$ and let $f(t)=v e^{t^{2}/2}$ then it suffices to prove

$J(v):=\int_{0}^{\infty} |\triangledown v|^{2}/2-v^{2}ln(|v|)dt\geq \frac{\sqrt{\pi}}{4}$

constrained to $\int_{0}^{\infty} v^{2}dt=\frac{\sqrt{\pi}}{2}$. But $\triangle v+2vln(v)+(\lambda+1)v=0$ ,where $\lambda$ is the lagrange multiplier, doesn't look easy to solve.

In the paper below, a different variational calculus argument is proposed, which I still don't understand.

http://www.jstor.org.myaccess.library.utoronto.ca/stable/pdf/2374139.pdf?acceptTC=true

Thank you