Sobolev embedding for weighted spaces

Question

Suppose that $ p \in (1,2) $, $ \tilde p = \frac{ 2p }{ 2 - p } $ and $ \rho > 1 $. Is the following result true: $$ W_{ \rho }^{ 1, p }(\mathbb{R}^2) \subset L^s_{ \rho }( \mathbb{R}^2 ) $$ for all $ s \in ( p, \tilde p ) $? Here $ L^s_{\rho}(\mathbb{R}^2) $ is the weighted Lebesgue space: $$ L^s_{\rho}(\mathbb{R}^2) = \{ f \in L^s(\mathbb{R}^2) : ( 1 + |\cdot|^2 )^{\rho/2} \, f(\cdot) \in L^s(\mathbb{R}^2) \} $$ and $ W^{1,p}_{\rho}(\mathbb{R}^2) $ is the corresponding weighted Sobolev space, i.e. the space of measurable functions with $ 1 $ weak derivative in $ L^{p}_{\rho}(\mathbb{R}^2) $. The symbol $ \subset $ stands for continuous embedding.

If it is true, is there a standard reference to this result?

Answer 1

A standard reference is "Sobolev spaces" from R.A. Adams and J.J.F. Fournier.

Answer 2

See https://terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/ Theorem 3 (Sobolev embedding theorem for one derivative)

The result is correct with or without weights.