Relations between different types of Hypoellipticity: A systematic approach

We give a systematic treatment to the concept of hypoellipticity, putting it into an abstract form which allows us to deal with several different notions within the same framework. We then investigate when a notion of hypoellipticity implies another one and, in particular, when it can be extended for more general spaces. We also present a relation between certain types of hypoellipticity and local solvability (for the transpose) for a family of operators.