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On a class of globally analytic Hypoelliptic operators with non-negative characteristic form

Nicholas Braun Rodrigues, Gregorio Chinni

TL;DR

The paper establishes global analytic hypoellipticity on the torus for a broad class of second-order operators with nonnegative characteristic form, extending the sum-of-squares theory of BC-2022. It develops a globalHOR framework via a basic estimate and subelliptic control for a structured operator $P(t,D_t,D_x)$ with block matrices $\mathbf{A}(t)$, formulating assumptions (A1)-(A3) to manage degeneracies. The main results are two theorems: Th1 shows global analyticity under (A1)-(A2) with $n_1=n$, and Th2 extends to the mixed regime under (A1)-(A2)+(A3), including operators not representable as sums of squares. The proofs combine microlocal Gevrey analysis, partition of unity, and Weierstrass preparation to obtain factorial growth bounds and conclude that the analytic wave front set is empty; the Appendix illustrates the breadth of the framework with explicit examples and non-Hörmander-type operators.

Abstract

The global analytic hypoellipticity is proved for a class of second order partial differential equations with non-negative characteristic form globally defined on the torus. The class considered in this work generalizes at some degree the class of sum of squares considered by Bove-Chinni and also by Cordaro-Himonas.

On a class of globally analytic Hypoelliptic operators with non-negative characteristic form

TL;DR

The paper establishes global analytic hypoellipticity on the torus for a broad class of second-order operators with nonnegative characteristic form, extending the sum-of-squares theory of BC-2022. It develops a globalHOR framework via a basic estimate and subelliptic control for a structured operator with block matrices , formulating assumptions (A1)-(A3) to manage degeneracies. The main results are two theorems: Th1 shows global analyticity under (A1)-(A2) with , and Th2 extends to the mixed regime under (A1)-(A2)+(A3), including operators not representable as sums of squares. The proofs combine microlocal Gevrey analysis, partition of unity, and Weierstrass preparation to obtain factorial growth bounds and conclude that the analytic wave front set is empty; the Appendix illustrates the breadth of the framework with explicit examples and non-Hörmander-type operators.

Abstract

The global analytic hypoellipticity is proved for a class of second order partial differential equations with non-negative characteristic form globally defined on the torus. The class considered in this work generalizes at some degree the class of sum of squares considered by Bove-Chinni and also by Cordaro-Himonas.

Paper Structure

This paper contains 8 sections, 12 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Notations, Definitions, Preliminary facts and Main Results
  3. Basic Estimate
  4. A class of globally analytic Hypoelliptic operatorst with non-negative quadratic form
  5. Proof of Theorems \ref{['Th_1']} and \ref{['Th_2']}
  6. Proof of the Theorem \ref{['Th_1']}
  7. Proof of the Theorem \ref{['Th_2']}
  8. Appendix(Some examples)

Key Result

Lemma 2.1

Let $P^{k}(x,D)$ be as in Pk_ud. Then there is a positive constant $C$ such that

Theorems & Definitions (29)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2: OR1971
  • Lemma 2.3
  • proof
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.2: (H.O.R.)-condition
  • Theorem 2.1
  • ...and 19 more