Global solvability and propagation of regularity of sums of squares on compact manifolds

Regularity and comparison principles in complex analysis and locally integrable structures

“…A much more involved model is the case of a product 𝑀 × 𝕋 𝑑 of a compact, complex manifold with the 𝑑-dimensional torus, where a natural CR Levi-flat structure can be defined after choosing suitable 𝑑-closed (0, 1)-forms on 𝑀. In this case, the associated real structure was studied by several authors (Bergamasco, P. D. Cordaro, and Malagutti, 1993, Bergamasco, P. D. Cordaro, and Petronilho, 1996, Araújo, Dattori da Silva, and Lessa Victor, 2022, Araújo, Ferra, Jahnke, and Ragognette, 2023. We introduce such models in Section 3 and describe some of their main properties.…”

“…Remark 1.6.2. We refer to the paper Araújo, Ferra, Jahnke, and Ragognette, 2023 for the definition of a weakly non-simultaneously approximable system.…”

“…We also compare properties of global hypoellipticity for both structures, and introduce a notion of global hypoellipticity for forms of higher degree. Using recent results on characterization of global hypoellipticity for  ⊕  (proved in Araújo, Dattori da Silva, andLessa Victor, 2022 andAraújo, Ferra, Jahnke, andRagognette, 2023), we are able to obtain characterizations of global hypoellipticity for 𝜕 𝑏 . Here, the main tool is the Laplacians associated to the natural metric on Ω and a Fourier decomposition of this (non-elliptic) operator into a countable sequence of elliptic operators.…”