TITLE: Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$
QUESTION [33 upvotes]: How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have $\lVert uv\rVert_s \leq C \lVert u\rVert_s \lVert v\rVert_s$. Recall that $\lVert f\rVert_s=\lVert(1+|\eta|^2)^{s/2}\,\hat{f}(\eta)\rVert$, the norm on $H^s(\mathbb{R}^n)$. This is an exercise from Taylor's book, Partial differential equations I.

REPLY [36 votes]: Note that
$$
\begin{split}
(1+|\xi|^2)^p
&\leq (1+2|\xi-\eta|^2+2|\eta|^2)^p\\
&\leq 2^p(1+|\xi-\eta|^2+1+|\eta|^2)^p\\
&\leq c(1+|\xi-\eta|^2)^p + c(1+|\eta|^2)^p,
\end{split}
$$
for $p>0$, where $c=\max\{2^{p},2^{2p-1}\}$. 
Put $\langle\xi\rangle=\sqrt{1+|\xi|^2}$.
Then we have
$$
\begin{split}
\langle\xi\rangle^s |\widehat{uv}(\xi)|
&\leq \int \langle\xi\rangle^s |\hat{u}(\xi-\eta)\hat{v}(\eta)|\,\mathrm{d}\eta\\
&\leq c\int \langle\xi-\eta\rangle^s |\hat{u}(\xi-\eta)\hat{v}(\eta)|\,\mathrm{d}\eta
+ c\int \langle\eta\rangle^s |\hat{u}(\xi-\eta)\hat{v}(\eta)|\,\mathrm{d}\eta\\
&\leq c|\langle\cdot\rangle^s\hat u|*|\hat v| + c|\hat u|*|\langle\cdot\rangle^s\hat v|,
\end{split}
$$
which, in light of Young's inequality, implies
$$
\|uv\|_{H^s} \leq c\|u\|_{H^s} \|\hat v\|_{L^1} + c\|\hat u\|_{L^1}\|v\|_{H^s}.
$$
Finally, we note that $\|\hat u\|_{L^1}\leq C\,\|u\|_{H^s}$ when $s>\frac{n}2$.