Show the convolution of a $C_c^\infty (\Bbb R^n)$ function with a $L^p(\Bbb R^n)$ function is in $C^\infty(\Bbb R^n)$, $1\le p\le\infty$ Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$.  Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$.
Let $x=(x_1,x_2,\ldots,x_n)$ and $y=(x_1+h,x_2,\ldots,x_n)$, $h \ne 0$.
The first step is to show
$$
\partial_{x_1} \left(f \ast g\right)= f \ast \left(\partial_{x_1}g\right),
$$
through the dominated convergence theorem.
Is it possible to bound
$$
\int_{\Bbb R^n} \left|f(t)\right| \frac{\left|g(x-t)-g(y-t)\right|}{|x-y|}dt
$$
with the maximal function of $g$?
Or resorting to the mean value theorem
$$
g(x-t)-g(y-t)=\int_0^1 Jg\left(x+t(y-x)\right)dt \cdot (y-x)
$$ 
is the only way, where $Jg$ is the Jacobian matrix of $g$.

Here is a proof without using dominated convergence: Let $e_1 = (1,0,\ldots,0) \in \mathbb{R}^n$, $x \in \mathbb{R}^n$, $t \in \mathbb{R}$. Then
$$(f \ast g)(x+t \cdot e_1)-(f \ast g)(x) = \int_{\mathbb{R}^n} (g(x+t \cdot e_1-y)-g(x-y)) \cdot f(y) \, dy \\ = \int_{\mathbb{R}^n} \int_0^t \partial_1 g(x+\tau \cdot e_1-y) \, d\tau f(y) \, dy \\ \stackrel{\text{Fubini}}{=} \int_0^t \int_{\mathbb{R}^n} \partial_1 g(x+t \cdot e_1-y) \cdot f(y) \, dy \, d\tau \\ = \int_0^t \underbrace{((\partial_1 g) \ast f)}_{\text{continuous}} (x+\tau \cdot e_1) \, d\tau$$
Thus (by First Fundamental Theorem of Calculus): $t \mapsto (f \ast g)(x+t \cdot e_1)$ is differentiable and (for $t=0$) we obtain $\partial_1 (f \ast g) = f \ast (\partial_1 g)$. 
