A unified proof of sharp bounds for the Jacobi heat kernel with trace and estimates of multiplicative constants

TL;DR

This work provides a unified, optimized proof of sharp, explicitly constant bounds for the Jacobi heat kernel across all parameter ranges, with detailed tracking of multiplicative constants. By introducing the Xi^{\alpha,\beta}_{\kappa} framework and leveraging a product formula for Jacobi polynomials, the authors connect Jacobi kernels to numerous related heat kernels, including spherical and compact rank one symmetric spaces, and produce quantitative large-time asymptotics. The method is organized into autonomous Steps A–F, enabling precise control of constants and potential future refinements, and yields practical implications for sharp bounds on kernels on CROSS spaces and beyond. The results supply explicit constants and time thresholds, enhancing applicability in analysis, probability, and mathematical physics where sharp heat-kernel estimates are essential.

Abstract

We give a unified and optimized proof of the sharp bounds for the Jacobi heat kernel, which were obtained gradually in several papers in recent years. We lay particular emphasis on tracing and estimating all constants appearing throughout the entire reasoning. This allows us to quantitatively control the multiplicative constants in the Jacobi heat kernel bounds in terms of the parameters involved. Consequently, analogous control extends to a number of interrelated heat kernels. In particular, we obtain quantitative control in terms of the associated dimension for the spherical heat kernel and for all other heat kernels on compact rank one symmetric spaces.

Theorems & Definitions (121)

• Theorem 1: NSSNSS2NSS3
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Lemma 2.1: NSS2