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On the distribution kernels of Toeplitz operators on CR manifolds

Chin-Yu Hsiao, Ood Shabtai

TL;DR

This work analyzes Toeplitz operators on compact CR manifolds and CR orbifolds, focusing on their distribution kernels. It proves a diagonal formula for the second coefficient in the kernel's symbol expansion, tying it to the principal symbol along the Reeb direction, Tanaka-Webster curvature, and subprincipal data, thereby extending Szegő kernel asymptotics to Toeplitz operators. The results bridge CR quantization with deformation-quantization notions by providing explicit expressions that yield CR star-product components, and they adapt the framework to orbifold settings via group averaging. Together, these results offer a precise, globally meaningful description of Toeplitz kernels and lay groundwork for CR deformation-quantization in both smooth and orbifold contexts.

Abstract

We study the distribution kernel of a Toeplitz operator associated with a classical pseudodifferential operator on a compact, embeddable, strictly pseudoconvex CR manifold. The main result consists of a formula for the values at the diagonal of the second coefficient in the expansion of the symbol of the kernel. We also establish asymptotic expansions for Toeplitz operators on the positive part of a compact not necessary strictly pseudoconvex CR orbifold under certain natural assumptions.

On the distribution kernels of Toeplitz operators on CR manifolds

TL;DR

This work analyzes Toeplitz operators on compact CR manifolds and CR orbifolds, focusing on their distribution kernels. It proves a diagonal formula for the second coefficient in the kernel's symbol expansion, tying it to the principal symbol along the Reeb direction, Tanaka-Webster curvature, and subprincipal data, thereby extending Szegő kernel asymptotics to Toeplitz operators. The results bridge CR quantization with deformation-quantization notions by providing explicit expressions that yield CR star-product components, and they adapt the framework to orbifold settings via group averaging. Together, these results offer a precise, globally meaningful description of Toeplitz kernels and lay groundwork for CR deformation-quantization in both smooth and orbifold contexts.

Abstract

We study the distribution kernel of a Toeplitz operator associated with a classical pseudodifferential operator on a compact, embeddable, strictly pseudoconvex CR manifold. The main result consists of a formula for the values at the diagonal of the second coefficient in the expansion of the symbol of the kernel. We also establish asymptotic expansions for Toeplitz operators on the positive part of a compact not necessary strictly pseudoconvex CR orbifold under certain natural assumptions.

Paper Structure

This paper contains 13 sections, 37 theorems, 397 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Deformation quantization of CR manifolds
  4. Preliminaries
  5. Notations
  6. Abstract CR manifolds and the Szegö kernel
  7. Local representations
  8. Some geometry on $T^*X$
  9. The expansion of the kernel of a Toeplitz operator
  10. Point-wise computations
  11. Expansions of products
  12. Distribution kernels represented by oscillatory integrals
  13. Distribution kernels of Toeplitz operators on CR orbifolds

Key Result

Theorem 1.1

Let $X$ be an orientable compact strongly pseudoconvex embeddable CR manifold of dimension $2n+1$, $n \ge 1$. Assume that $\mathbb C TX$ is equipped with the Levi metric (levi_metric). Let $(D,x)$ be a coordinate patch on $X$. Then where $\phi \in C^\infty(D \times D)$ satisfies and $a\in S^n_{\mathop{\mathrm{cl}}\nolimits}(D \times D \times \mathbb R_+)$ is a classical Hörmander symbol (see Def

Theorems & Definitions (70)

  • Theorem 1.1: bdms
  • Theorem 1.2: hsiao_shen
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 60 more