# 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.