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Sharp subelliptic estimates in the $\bar\partial$-Neumann problem via an uncertainty principle

Gian Maria Dall'Ara, Samuele Mongodi

Abstract

The problem of giving a (CR-)geometric description of the best possible order of a subelliptic estimate at a boundary point in the $\bar\partial$-Neumann problem is largely open. In this paper, we introduce a novel technique based on a "$\bar\partial$-uncertainty principle" and, as an application, we determine the sharp order of subellipticity at the origin for a large class of Kohn's special domains in ambient dimension $\leq 5$.

Sharp subelliptic estimates in the $\bar\partial$-Neumann problem via an uncertainty principle

Abstract

The problem of giving a (CR-)geometric description of the best possible order of a subelliptic estimate at a boundary point in the -Neumann problem is largely open. In this paper, we introduce a novel technique based on a "-uncertainty principle" and, as an application, we determine the sharp order of subellipticity at the origin for a large class of Kohn's special domains in ambient dimension .
Paper Structure (31 sections, 24 theorems, 180 equations)

This paper contains 31 sections, 24 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Subellipticity in the $\bar{\partial}$-Neumann problem
  3. The sharp order of subellipticity. Previous work
  4. Content of the paper
  5. Notation and terminology
  6. Acknowledgment
  7. Subellipticity via spectral gap estimates
  8. Preliminaries on pseudoconvex rigid domains
  9. A standard reduction to the boundary
  10. Two basic identities
  11. Fourier analysis
  12. Proof of Proposition \ref{['prp:spectral_gap']}
  13. An uncertainty principle for the $\overline{\partial}$ operator
  14. A basic $\overline{\partial}$-uncertainty principle
  15. A weighted $\overline{\partial}$-UP
  16. ...and 16 more sections

Key Result

Proposition 2.3

Let $s\leq 1$. Suppose that there exist a positive constant $C\geq 1$ such that the estimate holds for every $\xi\geq C$ and $u\in C_c^\infty(\{|z|<C^{-1}\})$. Then the $\overline{\partial}$-Neumann problem on the rigid domain satisfies a subelliptic estimate of order $s$ at the origin.

Theorems & Definitions (52)

  • Definition 1.1: Sharp order of subellipticity
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: Spectral gap estimates imply subellipticity
  • Proposition 2.4: $\mathbf{E}^{\varphi}$ behaves well under scalings
  • proof
  • Proposition 2.5: Reduction to a boundary subelliptic estimate
  • proof
  • Remark 2.6
  • Proposition 2.7: Basic identities
  • ...and 42 more