If I have a function $f \in H^s(\mathbb R^n)$, what can I say about $f^2$? In particular, I'm interested in the case when $s < 0$ and would like to conclude that, for example, $f \in H^{2s}(\mathbb R^n)$. Is something like this true? 



If a function is in a Sobolev space, what can we say about its square?
No, something like this is not true. In any negative Sobolev space, you can find a function f with a point singularity such that $f^2$ is not integrable. Hence $f^2$ is not even a distribution, and certainly not in $H^{2s}$.


