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Poisson and Szegö kernel scaling asymptotics on Grauert tube boundaries (after Zelditch, Chang and Rabinowitz)

Roberto Paoletti

Abstract

We review and elaborate on recent work of Chang and Rabinowitz on scaling asymptotics of Poisson and Szegö kernels on Grauert tubes, providing additional results that may be useful in applications. In particular, focusing on the near-diagonal case, we give an explicit description of the leading order terms, and an estimate on the growth of the degree of certain polynomials describing the rescaled asymptotics. Furthermore, we allow rescaled asymptotics in a range $O\left(λ^{ε-1/2}\right)$ in all the variables involved, where $λ\rightarrow+\infty$ is the asymptotic parameter, rather than rescale according to Heisenberg type.

Poisson and Szegö kernel scaling asymptotics on Grauert tube boundaries (after Zelditch, Chang and Rabinowitz)

Abstract

We review and elaborate on recent work of Chang and Rabinowitz on scaling asymptotics of Poisson and Szegö kernels on Grauert tubes, providing additional results that may be useful in applications. In particular, focusing on the near-diagonal case, we give an explicit description of the leading order terms, and an estimate on the growth of the degree of certain polynomials describing the rescaled asymptotics. Furthermore, we allow rescaled asymptotics in a range in all the variables involved, where is the asymptotic parameter, rather than rescale according to Heisenberg type.
Paper Structure (37 sections, 53 theorems, 292 equations)

This paper contains 37 sections, 53 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. An example
  3. Preliminaries
  4. Notation
  5. Fourier transform.
  6. Densities and functions.
  7. Schwartz kernels.
  8. Induced vector fields.
  9. Riemannian and Kähler structures, I.
  10. Hamiltonian vector fields.
  11. Riemannian and Kähler structures, II.
  12. Riemannian distance function
  13. Reeb vector fields.
  14. Volume form.
  15. Geodesic flow.
  16. ...and 22 more sections

Key Result

Theorem 3

Suppose $\tau\in (0,\tau_{max})$, $x\in X^\tau$, and $\chi\in \mathcal{C}^\infty_c((-\epsilon,\epsilon))$ for some suitably small $\epsilon>0$. Then the following holds.

Theorems & Definitions (93)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Theorem 6
  • Remark 7
  • Corollary 8
  • Proposition 9
  • Corollary 10
  • ...and 83 more