Let us define the convolution of distributions $S*T \in \mathcal{D}'(\mathbb{R}^d)$ with $S\in \mathcal{D}'(\mathbb{R}^d)$ with compact support and $T \in \mathcal{D}'(\mathbb{R}^d)$ by
$(S*T)(f) = T(\tilde{S}*f)$ for all $f \in \mathcal{D}(\mathbb{R}^d)$ where $\tilde{S}(g) = S(\tilde{g})$ for $g \in \mathcal{D}(\mathbb{R}^d)$ and $\tilde{g}(x) = g(-x)$.
What do you think are reasonable properties of the convolution in general and do they satisfy the definition above?