I have a function $f(x)$ in the space $H^{2,s}(\mathbb{R}^3)$; have this limit sense $$\lim_{xy\to 0} f(x)$$ ? ($y$ is a fixed point) If i have $f$ in $H^2$ I can say that $$\lim_{xy\to 0} f(x)=f(y)$$ because $H^2(\mathbb{R}^3)\subset C(\mathbb{R}^3)$, but in the space $H^{2,s}$? In general I can say nothing, isn't it?
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Not quite sure. If you are in $H^{2s}(\mathbb R^3)$ you can embed in $W^{1,\frac{6}{1+2s}}(\mathbb R^3)$, and from there you can get into $C^{0,\alpha}({\mathbb R}^3)$ by Morrey's inequality, for all $\alpha,s\in (0,1)$ such that $s+\alpha=\frac12$. So, I would say there's plenty of space for you beneath $H^2(\mathbb R^3)$ to still get a continuous function in the end. 

