Uniform pointwise estimates for ultraspherical polynomials

TL;DR

This paper establishes pointwise bounds for Jacobi polynomials, which are crucial for spectral analysis on spheres and for proving sharp multiplier theorems for associated Laplacians.

Contribution

It provides uniform pointwise estimates for ultraspherical polynomials, advancing spectral analysis and harmonic analysis on spheres in arbitrary dimensions.

Findings

Derived bounds enable sharper spectral multiplier theorems.

Applied bounds to analyze Laplacians and sub-Laplacians on spheres.

Facilitated progress in harmonic analysis on high-dimensional spheres.

Abstract

We prove pointwise bounds for two-parameter families of Jacobi polynomials. Our bounds imply estimates for a class of functions arising from the spectral analysis of distinguished Laplacians and sub-Laplacians on the unit sphere in arbitrary dimension, and are instrumental in the proof of sharp multiplier theorems for those operators.