It is well known that $\|fg\|_s \lesssim \|f\|_{s_1} \|g\|_{s_2}$ for functions $f: {\mathbb R}^n \rightarrow {\mathbb R}$ under certain conditions on $s$, $s_1$, $s_2$ (i.e. $s_1$, $s_2 \geq s$ and $s \leq s_1+s_2 -n/2$). Here $\|\cdot\|_t$ is the usual norm for $H^t({\mathbb R}^n)$ and all indices are non-negative.

My question is: Does the result also hold if we replace ${\mathbb R}^n$ with a (not necessarily bounded) domain $\Omega \subset {\mathbb R}^n$? (I need a result which is not restricted to integers.)