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Riesz transforms associated with the Grushin operator with drift

Nishta Garg, Rahul Garg

Abstract

We consider the Grushin operator with drift which is symmetric with respect to a measure having exponential growth. For the corresponding Riesz transforms, we study strong-type $(p, p)$, $1 < p < \infty$, and weak-type $(1, 1)$ boundedness.

Riesz transforms associated with the Grushin operator with drift

Abstract

We consider the Grushin operator with drift which is symmetric with respect to a measure having exponential growth. For the corresponding Riesz transforms, we study strong-type , , and weak-type boundedness.
Paper Structure (14 sections, 13 theorems, 136 equations)

This paper contains 14 sections, 13 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Basic results
  3. Grushin operator
  4. The Heisenberg-Reiter group
  5. Relation between the Grushin operator and the sub-Laplacian
  6. Grushin operator with drift
  7. Rotation and Dilation
  8. Rotation
  9. Dilation
  10. Regularized Riesz transforms and a transference principle
  11. Regularized Riesz transforms
  12. A transference principle
  13. Proofs of Theorems \ref{['thm:p_bdd']} and \ref{['thm:1_bdd-positive']}
  14. Proof of Theorem \ref{['thm:1_bdd-negative']}

Key Result

Theorem 1.1

For any $1< p <\infty$ and $\alpha \in \mathbb{N}^n$ with $|\alpha| \geq 1$, the Riesz transform $R_\alpha = \nabla^\alpha (\Delta_\nu)^{-|\alpha|/2}$ is bounded on $L^p(\mathbb{R}^n, d\mu_\nu).$

Theorems & Definitions (22)

  • Theorem 1.1: Lohoué--Mustapha Lohoue-Mustapha-drift-2004
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Lohoué--Mustapha Lohoue-Mustapha-drift-2004
  • Theorem 2.2: Li--Sjögren Heisenberg_group_drift
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 12 more