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On $L^2$-boundedness of pseudo-multipliers associated to the Grushin operator

Sayan Bagchi, Rahul Garg

Abstract

In this article we define analogues of pseudo-differential operators associated to the joint functional calculus of the Grushin operator using their spectral resolution, and study Calderón--Vaillancourt-type theorems for these operators.

On $L^2$-boundedness of pseudo-multipliers associated to the Grushin operator

Abstract

In this article we define analogues of pseudo-differential operators associated to the joint functional calculus of the Grushin operator using their spectral resolution, and study Calderón--Vaillancourt-type theorems for these operators.
Paper Structure (21 sections, 28 theorems, 256 equations)

This paper contains 21 sections, 28 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Pseudo-differential operators
  3. Hermite pseudo-multipliers
  4. Grushin pseudo-multipliers
  5. Pseudo-multipliers for the joint functional calculus of the Grushin operator
  6. Methodology of the proof
  7. Organisation of the paper
  8. Notations and parameters
  9. Preliminaries and basic results
  10. Pseudo-multipliers and spectral theory
  11. Control distance for the Grushin operator
  12. Heat kernel for the Grushin operator
  13. Weighted Plancherel estimates
  14. Partition of unity
  15. --boundedness in the case of
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $m \in S^0_{\rho, \delta}$, with $0 \leq \delta \leq \rho \leq 1, \, \delta \neq 1$. Then the operator $m(x, D)$ extends to a bounded operator from $L^2(\mathbb{R}^{n})$ to itself.

Theorems & Definitions (60)

  • Theorem 1.1: Calderón--Vaillancourt
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 50 more