Global Hypoellipticity for First-Order Operators on Closed Smooth Manifolds

TL;DR

This paper investigates the conditions under which certain first-order operators on closed manifolds are globally hypoelliptic, revealing that classical conditions like Niremberg-Treves (P) are not always applicable.

Contribution

It provides necessary and sufficient conditions for global hypoellipticity of first-order operators on closed manifolds, especially in cases with variable separation, challenging existing classical criteria.

Findings

Necessary and sufficient conditions for hypoellipticity are established.

The classical (P) condition is shown to be neither necessary nor sufficient in some cases.

The results extend understanding of hypoellipticity for operators on closed manifolds.

Abstract

The main goal of this paper is to address global hypoellipticity issues for the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \in \mathbb{T} \times M$, $\mathbb{T}$ is the one-dimensional torus, $M$ is a closed manifold and $C(t,x,D_x)$ is a first order pseudo-differential operator on $M$, smoothly depending on the periodic variable $t$. In the case of separation of variables, namely, $C(t,x,D_x) = a(t)p(x,D_x)+ib(t)q(x,D_x)$, we give necessary and sufficient conditions for the global hypoellipticity of $L$. In particular, we show that, under suitable conditions, the famous (P) condition of Niremberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of $L$.