Let $W^{k}_{2,w_1}(\mathbb{R})$ be a weighted sobolev space with positive continuous weight function $w_1$ for the integrals of the function and its derivatives. Let $W^{k}_{2,w_{1,1}}(\mathbb{R}^2)$ be a weighted sobolev space with positive continuous weight function $w_{1,1}(x,y):=w_1(x)w_1(y)$. Is then $W^{k}_{2,w_1}(\mathbb{R})\otimes W^{k}_{2,w_1}(\mathbb{R}) $ dense in $W^{k}_{2,w_{1,1}}(\mathbb{R}^2)$ under these conditions on the weights $w_1$ and $w_{1,1}$?
In particular, is $C_0^{\infty}(\mathbb{R^2})$ dense in $W^{k}_{2,w_{1,1}}(\mathbb{R}^2)$?