Convolution of a distribution with a test function

For an integer $n\ge 1$, $u\in {\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ as in Distribution, and $\phi \in \mathcal{D}({\mathbb{R}}^n)$, define

$(u\ast \phi )(x)=⟨u(y),\phi (x-y)⟩\mathrm{.}$

The variable $y$ is the distribution variable: for fixed $x$, the smooth test $y\mapsto \phi (x-y)$ has compact support $x-\mathrm{supp}\phi$. The pairing is complex bilinear, with no conjugation; the displayed test, not the pairing convention itself, includes reflection of $\phi$.

More generally, if $u\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$, the same formula is defined on the open set $\{x:x-\mathrm{supp}\phi \subseteq \mathrm{\Omega }\}$. Openness follows by covering the compact set $x-\mathrm{supp}\phi$ by finitely many balls whose closures lie in $\mathrm{\Omega }$, giving a positive translation margin. If $\phi =0$, its support is empty and the formula defines zero everywhere. The subsequent smoothness theorem proves differentiability and its signs; the definition only pairs legitimate compactly supported tests.