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Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators

Valentina Casarino, Paolo Ciatti, Alessio Martini

Abstract

We study degenerate elliptic operators of Grushin type on the $d$-dimensional sphere, which are singular on a $k$-dimensional sphere for some $k < d$. For these operators we prove a spectral multiplier theorem of Mihlin-Hörmander type, which is optimal whenever $2k \leq d$, and a corresponding Bochner-Riesz summability result. The proof hinges on suitable weighted spectral cluster bounds, which in turn depend on precise estimates for ultraspherical polynomials.

Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators

Abstract

We study degenerate elliptic operators of Grushin type on the -dimensional sphere, which are singular on a -dimensional sphere for some . For these operators we prove a spectral multiplier theorem of Mihlin-Hörmander type, which is optimal whenever , and a corresponding Bochner-Riesz summability result. The proof hinges on suitable weighted spectral cluster bounds, which in turn depend on precise estimates for ultraspherical polynomials.

Paper Structure

This paper contains 21 sections, 16 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. An abstract multiplier theorem
  3. Spherical Laplacians and Grushin operators
  4. The Laplace--Beltrami operator on the unit sphere
  5. A family of commuting Laplacians and spherical Grushin operators
  6. Cylindrical coordinates
  7. The sub-Riemannian distance
  8. A complete system of joint eigenfunctions
  9. Spectral theory of the Laplace--Beltrami operator
  10. Joint eigenfunctions of Delta d and Delta d-1
  11. Joint eigenfunctions of Delta d,...,Delta k
  12. Riesz-type bounds
  13. Estimates for ultraspherical polynomials
  14. Weighted spectral cluster estimates
  15. The subelliptic regime
  16. ...and 6 more sections

Key Result

Theorem \oldthetheorem

Let $D = \max\{d,2k\}$ and $s > D/2$.

Theorems & Definitions (28)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 18 more