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Carleman estimates for higher step Grushin operators

Hendrik De Bie, Pan Lian

Abstract

The higher step Grushin operators $Δ_α$ are a family of sub-elliptic operators which degenerate on a sub-manifold of $\mathbb{R}^{n+m}$. This paper establishes Carleman-type inequalities for these operators. It is achieved by deriving a weighted $L^{p}-L^{q}$ estimate for the Grushin-harmonic projector. The crucial ingredient in the proof is the addition formula for Gegenbauer polynomials due to T. Koornwinder and Y. Xu. As a consequence, we obtain the strong unique continuation property for the Schrödinger operators $-Δ_α+V$ at points of the degeneracy manifold, where $V$ belongs to certain $ L^{r}_{\rm loc}(\mathbb{R}^{n+m})$.

Carleman estimates for higher step Grushin operators

Abstract

The higher step Grushin operators are a family of sub-elliptic operators which degenerate on a sub-manifold of . This paper establishes Carleman-type inequalities for these operators. It is achieved by deriving a weighted estimate for the Grushin-harmonic projector. The crucial ingredient in the proof is the addition formula for Gegenbauer polynomials due to T. Koornwinder and Y. Xu. As a consequence, we obtain the strong unique continuation property for the Schrödinger operators at points of the degeneracy manifold, where belongs to certain .
Paper Structure (16 sections, 30 theorems, 203 equations, 3 tables)

This paper contains 16 sections, 30 theorems, 203 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Grushin operator
  4. Polar coordinates
  5. Jacobi polynomials
  6. Spherical harmonics for the Grushin operator
  7. Orthogonal basis for Grushin-harmonics
  8. The orthogonal projection and addition formula
  9. The $\mathfrak{sl}_{2}$ triple and Fischer decomposition
  10. Bounds for the Grushin-harmonic projection
  11. $L^{1}-L^{\infty}$ estimates
  12. Weighted $L^{2}-L^{2}$ estimates
  13. Tighter estimates on $\mathbb{R}^{n+1}$
  14. Carleman estimates and unique continuation
  15. Carleman inequalities
  16. ...and 1 more sections

Key Result

Theorem 2.6

There exists a constant $C>0$ such that for all $x\in [-1, 1]$, all $\alpha,\beta\ge0$, and all non-negative integers $n$.

Theorems & Definitions (77)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.1
  • proof
  • ...and 67 more