Corollary 2

Let $k$ be a natural number. Let $f\in \mathscr{L}^1(\mathbf{R}^n, \mu)$ and $g\in \mathscr{C}^k(\mathbf{R}^n)$. Suppose that for every $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant k$, the function $\partial^{\alpha}g$ is bounded. Then the convolution product $f*g$ belong to $\mathscr{C}^k(\mathbf{R}^n)$ and for every $\alpha$ such that $|\alpha |\leqslant k$, one has $\partial^{\alpha}(f*g) =f*\partial^{\alpha}g$.

One can apply corollary 1 to the space $X =\mathbf{R}^n$, to Lebesgue measure and to the mapping $h$ defined by $(x, t)\mapsto f(x)g(t-x)$ from $\mathbf{R}^n\times \mathbf{R}^n$ into $\mathbf{C}$; indeed, for every $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant k$, one has the inequality

$|\partial^{\alpha}h(x, t)|\leqslant ($sup$_{|\beta|\leqslant k}$ sup$_{y\in\mathbf{R}^n}|\partial^{\beta}g(y)|)|f(x)|$

whose right-hand side is an integrable function on $\mathbf{R}^n$.