Relations between different types of Hypoellipticity: A systematic approach

TL;DR

This paper provides a systematic, abstract framework for understanding various notions of hypoellipticity, exploring their interrelations, extensions to broader spaces, and connections to local solvability of operators.

Contribution

It introduces a unified abstract approach to hypoellipticity, analyzing implications among different types and extending results to general spaces.

Findings

Established conditions under which one hypoellipticity notion implies another

Extended hypoellipticity concepts to more general functional spaces

Linked hypoellipticity types to local solvability for operator transposes

Abstract

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.