Local and Non-local Convolution Operators

Let K be a linear operator mapping a linear set X ⊂ D’(ℝd) into D’(ℝd). We denote X and K(X) by dom K and im K (respectively). The operator K is called local if does not increase the support, i.e. (Loc)$$T\; \in \;dom\;k\; \Rightarrow \;\operatorname{supp} \;k\left( T \right)\; \subset \;\operatorname{supp} \;T $$ In other words, for any open O ⊂ℝd the distribution K (T)|O depends only on the restriction of T onto the same set O: (1)$$\${T_{1,}}{T_2}\; \in \;dom\;K,\;{T_1}|0 = {T_2}|0 \Rightarrow \;K\left( {{T_1}} \right)|0 = K\left( {{T_2}} \right)|0$$.