I am interested in a version of Morrey's inequality but for a weighted Sobolev space. The classical Morrey inequality states, Let $p>N$, then for all $u \in W^{1,p}(\mathbb{R}^N)$,
$$ |u(x)-u(y)|\leq C|x-y|^{\alpha} \cdot\|\nabla u\|_{L_p}, $$ with $\alpha=1-\frac{N}{p}$. My question is does a similar result hold if we are working in a weighted Sobolev space, i.e let,
$$ W_\omega^{1,p}(\mathbb{R}^N)=\{ u:\int_\mathbb{R}|u|^p\omega_1dx + \int_{\mathbb{R}}|\nabla u|^p\omega_2dx < \infty \}, $$ with $\omega_1$ and $\omega_2$ are positive weight functions. I have tried to prove a result like this on my own but with no avail, approximating the function $u$ by it's derivative becomes more delicate. Any references would be greatly appreciated.