Geometrical proofs for the global solvability of systems

Abstract

We study a linear operator associated with a closed non‐exact 1‐form b defined on a smooth closed orientable surface M of genus g>1. Here we present two proofs that reveal the interplay between the global solvability of the operator and the global topology of the surface. The first result brings an answer for the global solvability when the system is defined by a generic Morse 1‐form. Necessary conditions for the global solvability bearing on the sublevel and superlevel sets of primitives of a smooth 1‐form b have already been established; we also present a more intuitive proof of this result.

Globally hypoelliptic triangularizable systems of periodic pseudo‐differential operators

“…Investigations on pseudo-differential classes are also considered in the literature, for instance, [10,[13][14][15]28]. These problems are also analyzed in the setting of compact manifolds and Lie groups, for example, [1,2,9,22,25]. In particular, we point out that an important tool, present in all these references and here, is a Fourier analysis characterizing the functional spaces under investigation.…”