Does convolution of two functions in $H^s(\mathbb{R})$ belong to $H^{2s}(\mathbb{R})$?

Question

Let $f$, $g$ be two density functions and assume that $f,g\in H^s(\mathbb{R})$, $s>\frac{1}{2}$, where $${H^s}(\mathbb{R}) = \left\{ {u:\int_{ - \infty }^\infty {{{\left| {\hat u\left( t \right)} \right|}^2}{{\left( {1 + {t^2}} \right)}^s}dt} < \infty } \right\}.$$ Does $f*g$ belong to $H^{2s}(\mathbb{R})$? Here the symbol $*$ denotes convolution, and $\hat u$ is the Fourier transform of $u$.

Answer 1

Build a pair of functions $u,v \in L^2(\mathbb{R})$ such that $u^2v^2\notin L^1(\mathbb{R})$ and put $\hat{f}=u/(1+|\xi|^2)^{s/2}$, $\hat{g}=v/(1+|\xi|^2)^{s/2}$.