convolution of Schwartz functions with $f(x) = (1+\|x\|)^{-\frac{1}{2}}$

Question

Let $f(x) = (1+\|x\|)^{-\frac{1}{2}}$ for $x \in \mathbb{R}^n$. This is clear that $f\star g \notin \mathcal S$ where $\mathcal S$ is algebra of Schwartz functions on $\mathbb{R}^n$ and $ g \in \mathcal S$. My question: does $f\star g$ belong to Schatten ideal $ S^ {P}$ for some $p$?