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I define the following wheighted 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 $$H^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg| D^\alpha u\in L^{2,s}(\mathbb{R}^3),|\alpha|\leq 2\bigg\rbrace$$ 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?

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Continuity is a local property: functions which are in your $L^{2,s}$ are locally in $L^2$, so functions in $H^{2,s}$ are locally in the Sobolev space $H^2(\mathbb R^3)$, thus are continuous functions (even Hölder 1/2-$\epsilon$).

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  • $\begingroup$ For negative s too? $\endgroup$ – Sue Mar 13 '13 at 13:50
  • $\begingroup$ Yes of course, because of the $(1+\vert x\vert^2)^s$, locally in $x$ comparable to 1. $\endgroup$ – Bazin Mar 13 '13 at 15:34
  • $\begingroup$ So the limit $$\lim_{|x-y|\to 0}f(x)$$, with $y\in\mathbb{R}^3$, has sense for $f\in H^{2,s}$. $\endgroup$ – Sue Mar 13 '13 at 15:39

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