Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global hypoellipticity for perturbations of complex vector fields on the torus

Maria V. Bartmeyer, Paulo L. Dattori da Silva, Rafael B. Gonzalez

Abstract

We apply Krönecker's approximation theorem to measure (in a topological sense) a set of constants which turn a vector field into a non-globally hypoelliptic operator. We present situations in which this set is a discrete enumerable (hence, meager) subset of the real line, and we also show that this set may be a dense $\mathcal{G}_δ$ subset of the complex numbers (hence, nonmeager), which produces a contrast to a known result stating that this set has null Lebesgue measure.

Global hypoellipticity for perturbations of complex vector fields on the torus

Abstract

We apply Krönecker's approximation theorem to measure (in a topological sense) a set of constants which turn a vector field into a non-globally hypoelliptic operator. We present situations in which this set is a discrete enumerable (hence, meager) subset of the real line, and we also show that this set may be a dense subset of the complex numbers (hence, nonmeager), which produces a contrast to a known result stating that this set has null Lebesgue measure.
Paper Structure (3 sections, 11 theorems, 78 equations)

This paper contains 3 sections, 11 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications: GH for operators with variable coefficients

Key Result

Theorem 1.1

Given $\alpha\in\mathbb{C},$ we have $\mathcal{M}_1=\mathcal{N}_1$ and

Theorems & Definitions (23)

  • Theorem 1.1: see B
  • Remark 1.2
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 13 more