Embedding of Sobolev spaces

Question

I define the following weighted Sobolev spaces $$L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$$ and $$H^{2,s}(\mathbb{R}^3)=\left\lbrace u\bigg|\,D^\alpha u\in L^{2,s}(\mathbb{R}^3),\,|\alpha|\leq 2\right\rbrace$$ I know that the classical Sobolev space $H^2(\mathbb{R}^3)$ is contained in $C(\mathbb{R}^3)$ because $2>\frac{3}{2}$. Can I extend this result to the above weighted Sobolev spaces?

Answer 1

In the comments you've already solved the "local" case of the problem.

For the "global" case where you give $C(\mathbb{R}^3)$ the uniform norm, the result is false, at least for sufficiently bad $s$. Simply consider the function $u(x) = (1 + |x|^2)^{t/2}$ where $t + s < -3/2$ and $t > 0$. This is possible if $s < -3/2$. Then you have that $u \in H^{2,s}$ but $u \not\in L^\infty$.