Uniform pointwise estimates for ultraspherical polynomials

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.

12 pages. This work extends the estimates for spherical harmonics initially proved in arXiv:1705.07068

References in corpus (6)

• Jacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator
• Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
• From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere
• Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces
• A robust approach to sharp multiplier theorems for Grushin operators
• Almost everywhere convergence of Bochner-Riesz means on Heisenberg-type groups