Let $ n \in \mathbb{N} $ and $ p \in \mathbb{R}_{\geq 1} $. If $ f \in \mathscr{S}(\mathbb{R}^{n}) $ and $ g \in {L^{p}}(\mathbb{R}^{n}) $, then it is a well-known fact from real analysis that the convolution $ f \star g $ is defined almost everywhere on $ \mathbb{R}^{n} $ and that $$ \| f \star g \|_{p} \leq \| f \|_{1} \| g \|_{p}. \quad (\text{Young's Inequality}) $$ This implies that $ f \star g \in {L^{p}}(\mathbb{R}^{n}) $.
My question: Is it true that $ f \star g $ is a smooth $ L^{p} $-function?
Thank you very much for your help!