If $f$ is a bounded tempered distribution and $g \in L^1$ is then $\int_{\Bbb R^n}(f\ast\tilde\varphi)(x)\tilde g(x)\,dx$ a tempered distribution? Let $f$ be a bounded tempered distribution, that is, $f\ast\varphi \in L^\infty(\mathbb R^n) $ for every Schwartz function $\varphi$. If $g \in L^1(\mathbb R^n)$, does the following definition define a tempered distribution: $$ \langle f \ast g,\varphi \rangle = \int\limits_{{\mathbb R^n}} {(f \ast \tilde \varphi } )(x)\tilde g(x) \, dx$$ where $\varphi$ is a Schwartz function and $\tilde \varphi(x)=\varphi(-x)$?

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}| \
onumber \\\ &\leq& \|f\ast \tilde{\phi}\|_{\infty}\|\tilde{g}\|_{1} \
onumber \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}| \
onumber \\\ &\leq& Cp_{k}(\tilde{\phi})\|\tilde{g}\|_{1} \
onumber \end{eqnarray}

With the last inequality you can conclude.