Let $n\in \mathbb N^*$ and let $s>n/2$. Then the Sobolev space $H^s(\mathbb R^n)$ is a multiplicative algebra included in $L^\infty(\mathbb R^n)$ and we have $ \Vert uv\Vert_{H^s(\mathbb R^n)}\le C(s,n)\Vert u\Vert_{H^s(\mathbb R^n)}\Vert v\Vert_{H^s(\mathbb R^n)}. $
I believe that we also have $$ \Vert uv\Vert_{H^s(\mathbb R^n)}\le C'(s,n)\bigl(\Vert u\Vert_{L^\infty(\mathbb R^n)}\Vert v\Vert_{H^s(\mathbb R^n)} +\Vert v\Vert_{L^\infty(\mathbb R^n)}\Vert u\Vert_{H^s(\mathbb R^n)}\bigr). $$ I found an answer in J.-Y. Chemin book MR1688875, Corollary 2.4.1.