# From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

## TL;DR

This paper establishes a sharp Mihlin-H"ormander type multiplier theorem for the Grushin operator on the sphere, using precise bounds for spherical harmonics, and proves boundedness of related Bochner-Riesz means.

## Contribution

It introduces a new sharp multiplier theorem for the Grushin operator on the sphere, advancing harmonic analysis in this setting.

## Key findings

- Proved a sharp Mihlin-H"ormander multiplier theorem for the Grushin operator.
- Established boundedness of Bochner-Riesz means associated with the operator.
- Derived precise pointwise bounds for spherical harmonics.

## Abstract

We prove a sharp multiplier theorem of Mihlin-H\"ormander type for the Grushin operator on the unit sphere in $\mathbb{R}^3$, and a corresponding boundedness result for the associated Bochner-Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics.