Yes. Take a look on this book in the part of tempered distributions:
"Michael Eugene Taylor - Partial Differential Equations Volume I Basic Theory"
Ok lets try. Let $S$ be the space of Schwartz functions.
Note first that this number is well defined:
\begin{eqnarray} |\langle f\ast g,\phi\rangle| &\leq& \int_{\mathbb{R}^{n}}|f\ast \tilde{\phi}||\tilde{g}| \nonumber \\ &\leq& \|f\ast \tilde{\phi}\|_{\infty}\|\tilde{g}\|_{1} \nonumber \end{eqnarray}
On the other hand, as you can see in that book, $f\ast \tilde{\phi}$ is a tempered distribution, so its is continuous i.e. $(\forall\phi)\Bigl(\phi\in S\Rightarrow |f\ast \tilde{\phi}|\leq Cp_{k}(\tilde{\phi})\Bigr)$ where $p_{k}(\phi)$ is defined as there "in the book".
Now
\begin{eqnarray} |\langle f\ast g,\phi\rangle| &\leq& \int_{\mathbb{R}^{n}}|f\ast \tilde{\phi}||\tilde{g}| \nonumber \\ &\leq& Cp_{k}(\tilde{\phi})\|\tilde{g}\|_{1} \nonumber \end{eqnarray}
With the last inequality you can conclude.