So, $C^{-\infty}(\Omega) := (C_c^{\infty}(\Omega))'$, that is the space of all continuous linear functionals over the space of compactly supported smooth functions. $C^{-\infty}(\mathbb{R}^n)$ is larger than tempered distributions $S'(\mathbb{R}^n)$ and we would get elements of $C^{-\infty}$ that aren't in $S'$ if we took for example the integral pairing, $ u \Psi \to \int \phi \Psi$ for some $\phi \in C^{\infty}$.
My question is what is the motivation behind these very general distributions? Are there examples where being in tempered distributions is not general enough? We have for example that Fourier transform is an isomorphism on $S'$ which is pretty useful but we don't have nice properties (I should say that I don't know any) on the very general $C^{-\infty}$, so I am wondering where they are needed. Thank you!
Speaking from observation but not experience, it seems people mainly consider tempered distributions in settings that make extensive use of Fourier analysis.
Linear functionals on $C_c(\Omega)$ called Radon measures. Off hand, I can think of two reasons why one might like to consider them:
Generalizing further to linear functionals on $C_c^{\infty}(\Omega)$, I presume, is something you'd do when you're interested in derivatives.