Products of functions in fractional-order Sobolev spaces

Question

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.)

Answer 1

Let $n\ge 1$ be an integer and $s>n/2$. Then you have $H^s(\mathbb R^n)\subset L^\infty(\mathbb R^n)$ and for $f,g\in H^s(\mathbb R^n)$, $$ \Vert fg \Vert_{H^s(\mathbb R^n)}\le c_n\bigl(\Vert f \Vert_{L^\infty(\mathbb R^n)} \Vert g \Vert_{H^s(\mathbb R^n)} +\Vert f \Vert_{H^s(\mathbb R^n)}\Vert g \Vert_{L^\infty(\mathbb R^n)}\bigr). $$ Of course the above estimate can be localized. If $\Omega$ is an open subset of $\mathbb R^n$, $s>n/2$ and $f,g\in H^s_{\text{loc}}(\Omega)$, $\chi, \psi\in C^\infty_c(\Omega)$, you get $$ \Vert \chi f\psi g \Vert_{H^s(\mathbb R^n)}\le c_n\bigl(\Vert \chi f \Vert_{L^\infty(\mathbb R^n)} \Vert \psi g \Vert_{H^s(\mathbb R^n)} +\Vert \chi f \Vert_{H^s(\mathbb R^n)}\Vert \psi g \Vert_{L^\infty(\mathbb R^n)}\bigr), $$ and this proves that $fg\in H^s_{\text{loc}}(\Omega)\subset L^\infty_{\text{loc}}(\Omega)$.