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Geometric results for hyperbolic operators with spectral transition of the Hamilton map

Enrico Bernardi, Tatsuo Nishitani

TL;DR

This work develops a comprehensive microlocal framework for a class of second-order non-effectively hyperbolic operators with a spectral transition of the Hamilton map along the doubly characteristic set $Σ$. By deriving a general normal form in symplectic coordinates and proving an extension lemma for functions on $Σ'$, the authors relate the Hamilton flow to the transition via an invariant $θ$ that encodes hyperbolicity and tangency phenomena. They establish criteria for the existence of tangent bicharacteristics and provide conditions for microlocal elementary factorization, a key step towards weighted energy estimates in a suitable calculus. The results generalize transition scenarios to higher codimension doubly characteristic manifolds and lay the groundwork for well-posedness and energy estimates in the presence of Hamilton-map spectral transitions and potential Gevrey-type thresholds.

Abstract

In this paper we study a class of non-effectively hyperbolic operators vanishing of order 2 on a manifold, on a sub-region of which the spectral structure of the Hamilton map changes type. Suitable normal symplectic coordinates are found together with an analysis of the Hamilton system associated to the principal symbol and a factorization result, preparing the operator for a microlocal energy estimate, is finally proven.

Geometric results for hyperbolic operators with spectral transition of the Hamilton map

TL;DR

This work develops a comprehensive microlocal framework for a class of second-order non-effectively hyperbolic operators with a spectral transition of the Hamilton map along the doubly characteristic set . By deriving a general normal form in symplectic coordinates and proving an extension lemma for functions on , the authors relate the Hamilton flow to the transition via an invariant that encodes hyperbolicity and tangency phenomena. They establish criteria for the existence of tangent bicharacteristics and provide conditions for microlocal elementary factorization, a key step towards weighted energy estimates in a suitable calculus. The results generalize transition scenarios to higher codimension doubly characteristic manifolds and lay the groundwork for well-posedness and energy estimates in the presence of Hamilton-map spectral transitions and potential Gevrey-type thresholds.

Abstract

In this paper we study a class of non-effectively hyperbolic operators vanishing of order 2 on a manifold, on a sub-region of which the spectral structure of the Hamilton map changes type. Suitable normal symplectic coordinates are found together with an analysis of the Hamilton system associated to the principal symbol and a factorization result, preparing the operator for a microlocal energy estimate, is finally proven.

Paper Structure

This paper contains 12 sections, 18 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Normal form
  3. Some notations
  4. A normal form
  5. Extension lemma
  6. Extension of $C^{\infty}(\Sigma')$
  7. More about the extension
  8. Tangent bicharacteristics
  9. Case that $d\theta$, $d\phi_j$ are linearly independent
  10. Case that $d\theta$, $d\phi_j$ are linearly dependent
  11. Elementary factorization
  12. Elementary factorization

Key Result

Lemma 2.1

Assume eq:rank. We have $W(\bar{\rho})\neq \{0\}$ if the spectral transition occurs at $\bar{\rho}$.

Theorems & Definitions (35)

  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Lemma 2.3
  • proof
  • Remark 2.2
  • Lemma 3.1
  • ...and 25 more