On the whole space $\mathbb{R}^d$ consider the weight $\omega(x)=\sqrt{1+|x|}$. Under which conditions on $k,q$ is the embedding $$ L^p(\mathbb{R}^d,\omega(x)dx)\subset\subset (W^{k,q}(\mathbb{R}^d,\omega(x)dx))^{\prime} $$ compact ($p$ is given)?

I am writing $(W^{k,q}(\mathbb{R}^d,\Omega(x)dx))^{\prime}$ because I am not too sure about the notation $(W^{k,q})'=W^{-k,q'}$ for such **weighted** spaces...