We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. We are given an $f \in L^p$ with $1\leqslant p < \infty$ and $g \in \mathcal{S}$. We want to show that $f \star g \in \mathcal{S}$, where $\star$ denotes the convolution operator.

I have already shown that $f \star g \in C^\infty$ by proving that $\partial^\alpha (f \star g) = f \star (\partial^\alpha g)$. Now I need to show that $(1+|x|^m)|\partial^\alpha(f \star g)(x) = (1+|x|^m)|f \star (\partial^\alpha g)(x)|$ is bounded. Since $\mathcal{S}$ is closed under differentiation, it suffices to consider $\alpha = 0$. I write $$ \int_{\mathbb{R}^n}f(y)g(x-y)(1+|x|^m)dy $$ and try to bound it but can't seem to make it work out. Could anyone help me proceed?