Proposition: The affine extension space

Let $t^{\circ }\in {\mathcal{D}}^{\prime }(U_{\mathrm{can}};V)$. If one extension $T_0\in {\mathcal{D}}^{\prime }(W;V)$ exists, all extensions are exactly

where ${{\mathcal{D}}^{\prime }}_Z$ denotes distributions supported in $Z$. This statement does not assert existence of an extension.

Proof. Two extensions have difference zero on $U_{\mathrm{can}}$, equivalently difference supported in $Z$. Conversely, adding a distribution supported in $Z$ does not change the restriction.