Global solvability and hypoellipticity for evolution operators on tori and spheres

TL;DR

This paper investigates the global solvability and hypoellipticity of a class of evolution operators on products of tori and spheres, providing new and existing results through Fourier analysis and Diophantine conditions.

Contribution

It extends known results and introduces new conditions for solvability and hypoellipticity of evolution operators on complex manifolds using Fourier and Diophantine analysis.

Findings

Necessary and sufficient conditions for solvability and hypoellipticity

Recovery of known results in special cases

New criteria involving Diophantine inequalities and level set connectivity

Abstract

In this paper, we study the global properties of a class of evolution-like differential operator with a 0-order perturbation defined on the product of $r+1$ tori and $s$ spheres $\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s$, with $r$ and $s$ non-negative integers. By varying the values of $r$ and $s$, we show that it is possible to recover results already known in the literature and present new results. The main tool used in this study is Fourier analysis, taken partially with respect to each copy of the torus and sphere. We obtain necessary and sufficient conditions related to Diophantine inequalities, change of sign and connectivity of level sets associated the operator's coefficients.