For every $s\in \mathbb{R}$, let us define $$H^{s}(\mathbb{R}^n)=\{u\in S^{\prime}(\mathbb{R}^n) :(1+|\xi|^{2})^{s/2}\hat{u}\in L^{2}(\mathbb{R}^n)\},$$ where $S^{\prime}(\mathbb{R}^n)$ is the space of all tempered distributions on $\mathbb{R}^{n}$.
I want to prove that if $u\in H^{s}(\mathbb{R}^{n})$ and $\phi\in S(\mathbb{R}^{n})$ (space of rapidly decreasing functions), then $\phi u \in H^{s}(\mathbb{R}^{n})$.
I know that I should apply an approximation argument (because of the density of $S(\mathbb{R}^{n})$ in $H^{s}(\mathbb{R}^{n})$). So, suppose that $u\in S(\mathbb{R}^{n})$. Then, $\widehat{(u\phi)}=\hat{u}*\hat{\phi}$.
How can I proceed in order to show that $(1+|\xi|^{2})^{s/2}(\hat{u}*\hat{\phi})\in L^{2}(\mathbb{R}^n)$?
Than you.
They key ingredient is Peetre's inequality, which states that for any $s \in \mathbb R$ and $\xi, \eta \in \mathbb R^n,$ we have,
$$ (1+|\xi|^2)^s \leq 2^{|s|} (1+|\xi-\eta|^2)^{|s|}(|1+|\eta|^2)^s. $$
I will omit the details of the proof, but the idea is to note that $1+|\xi+\eta|^2 \leq 2(1+|\xi|^2)(1+|\eta|^2)$ and take powers, treating the cases $s>0$ and $s<0$ separately.
We will also use Young's convolution inequality, which states that for $f \in L^p(\mathbb R^n), g \in L^q(\mathbb R^n)$ such that $\frac1p+\frac1q=\frac1r+1,$
$$ \lVert f \ast g \rVert_{L^r(\mathbb R^n)} \leq \lVert f \rVert_{L^p(\mathbb R^n)} \lVert g \rVert_{L^q(\mathbb R^n)}. $$
Using Peetre, we obtain a pointwise estimate,
\begin{align} (1+|\xi|^2)^{s/2} |(\hat u \ast \hat \phi)(\xi)| &\leq \int_{\mathbb R^n} (1+|\xi|^2)^{s/2} |\hat u(\eta)||\hat\phi(\xi-\eta)| \,\mathrm{d}\eta \\ &\leq 2^{|s|/2} \int_{\mathbb R^n}\left((1+|\eta|^2)^{s/2}\hat u(\eta)\right)\left((1+|\xi-\eta|^2)^{|s|/2}\hat\phi(\xi-\eta)\right) \,\mathrm{d}\eta \\ &\leq 2^{|s|/2} ((1+|\cdot|^2)^{s/2}\hat u) \ast ((1+|\cdot|^2)^{|s|/2}\hat \phi)(\xi) \end{align}
Hence by Young,
\begin{align} \lVert (1+|\xi|^2)^{s/2}(\hat u \ast \hat \phi)(\xi)\rVert_{L^2(\mathbb R^n)} &\leq 2^{|s|/2} \lVert (1+|\xi|^2)^{s/2}\hat u(\xi)\rVert_{L^2(\mathbb R^n)}\lVert(1+|\xi|^2)^{|s|/2}\hat \phi(\xi)\rVert_{L^2(\mathbb R^n)}, \end{align}
which establishes the required estimate, by noting $\mathcal{S}(\mathbb R^n) \subset H^{|s|}(\mathbb R^n).$