functional analysis – Compact Embedding of Weighted Sobolev space $W^{2,1}( rho dx)$ in $L^q( rho dx)$

$rho$ – continuous positive probability density on $mathbb{R}^n$.
Are $W^{2,1}( rho dx) subset L^1( rho dx)$ and $W^{2,1}( rho dx) subset L^2( rho dx)$ compact embedding?

For q > 2 $ W^{2,1}(gamma)notsubset L^{q}(gamma) $ where $gamma$ is standart normal distribution on $mathbb{R}$.

In both cases $L^1( rho dx)$ and $L^2( rho dx)$there are continuous embedding.