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

TL;DR

This paper establishes optimal spectral multiplier theorems and Bochner-Riesz summability results for degenerate elliptic Grushin-type operators on spheres, using weighted spectral cluster bounds and ultraspherical polynomial estimates.

Contribution

It introduces sharp spectral bounds and a Mihlin-H"ormander type multiplier theorem for ultraspherical Grushin operators on spheres, extending spectral analysis in degenerate elliptic settings.

Findings

Proves a spectral multiplier theorem of Mihlin-H"ormander type for these operators.

Establishes optimal bounds when 2k ≤ d.

Provides Bochner-Riesz summability results for the operators.

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-H\"ormander 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.