Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure (weight) $\gamma_n(x) dx$ and consequently $L^2(\mathbb{R}^n,d\gamma_n)$ and $H^1(\mathbb{R}^n,d\gamma_n)$ be the weighted versions of the Sobolev spaces with respect to the measure $d\gamma_n$.
Does the following weighted version of the Rellich-Kondrachov Theorem hold?
$$ H^1(\mathbb{R}^n,d\gamma_n) \text{ is } \textbf{compactly} \text{ embedded in } L^2(\mathbb{R}^n,d\gamma_n). $$