My definition for "weighted Sobolev space" $W^{1,p} _w (\Omega)$ is just a set of functions with the property that they admit distributional derivatives and that
$$ \int_\Omega |f|^p (x) w(x) dx \text{ and } \int_{\Omega} |\nabla f|^p w(x) dx$$
are both finite. Now, you can assume $w(x)$ to be, let's say, positive and bounded, I don't care too much about the generality of the weight for the moment. Also, you can assume $\Omega$ to be as regular as needed. What I am looking for are some references where to find proofs of classical inequalities (Poincaré-Wirtinger, Sobolev, Gagliardo-Niremberg interpolation inequality in particular etc...) in the context of weighted Sobolev spaces, everything I find seems to cover only the classical case with the Lebesgue measure.
