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Analysis of the critical CR GJMS operator

Yuya Takeuchi

Abstract

The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a non-hypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR $Q$-curvature.

Analysis of the critical CR GJMS operator

Abstract

The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a non-hypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR -curvature.

Paper Structure

This paper contains 5 sections, 19 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. CR manifolds
  3. Model operators on the Heisenberg group
  4. Heisenberg calculus
  5. Proofs of the main results

Key Result

Theorem 1.1

The operator $P$ is self-adjoint and has closed range.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • Definition 3.2
  • proof : Proof of \ref{['prop:well-defined-of-Hpsido']}
  • Theorem 3.3
  • ...and 30 more