Suppose that $s\geqslant 0$. Prove that there exists a constant $C$ such that $\forall u,v\in H^s(\mathbb R^n)\cap L^\infty(\mathbb R^n)$, one has the estimate $$ \lVert uv\rVert_{H^s(\mathbb R^n)}\leqslant C(\lVert u\rVert_{H^s(\mathbb R^n)}\lVert v\rVert_{L^\infty(\mathbb R^n)}+\lVert u\rVert_{L^\infty(\mathbb R^n)}\lVert v\rVert_{H^s(\mathbb R^n)}). $$ This is also known as Kato-Ponce inequality and I've already found some papers relevant to it, but I'm wondering that if there exists some elementary approaches. Note that we don't have $s>\frac n2$. Much thanks!
Edit: The Sobolev norm is defined as $$\lVert u\rVert_{H^s(\mathbb R^n)}:=\lVert (1+|\xi|^2)^{\frac s2}\hat u(\xi)\rVert_{L^2(\mathbb R^n)}=\left(\int_{\mathbb R^n}(1+|\xi|^2)^{s}|\hat u(\xi)|^2\mathrm d\xi\right)^{\frac12},$$ where $\hat u$ denotes the Fourier Transform of $u$.
