Relations between different types of Hypoellipticity: A systematic approach
Bruno de Lessa Victor, Luis F. Ragognette
TL;DR
This work develops an abstract, sheaf-theoretic framework for hypoellipticity that unifies multiple regularity notions across spaces such as $\mathcal{D}'$, $\mathcal{B}$, $\mathcal{C}^{\omega}$, $\mathcal{C}^{\infty}$, and Gevrey/ultradistribution scales. It proves that, for constant-coefficient operators in Gevrey contexts, many hypoellipticity notions are equivalent, and that commutativity with elliptic operators enables lifting regularity across Sobolev scales. It further analyzes extendability to ultradistributions, showing how tube-type and product-space operators admit Gevrey-to-ultradistribution extensions under precise hypotheses, while also clarifying the behavior of transposed operators (the hypoellipticity of $P$ implying solvability properties for $P^{\mathrm{t}}$) in the Gevrey, analytic, and hyperfunction settings. The paper also develops Gevrey hypoellipticity for systems of vector fields and establishes links between local solvability and transposed hypoellipticity via Holmgren-type results and delta-type considerations. Overall, the results provide a comprehensive map of how different hypoellipticity notions relate, extend, or fail across function spaces, with concrete examples (e.g., Lewy, Mizohata, Baouendi–Goulaouic) illustrating sharp thresholds and limitations.
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.