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

TL;DR

This paper provides a unified, optimized proof for sharp bounds of the Jacobi heat kernel, emphasizing precise constant estimation, which extends to related heat kernels on symmetric spaces and improves quantitative control.

Contribution

It offers a comprehensive proof that tightly bounds the Jacobi heat kernel with explicit constants, enhancing understanding of related heat kernels on symmetric spaces.

Findings

Sharp bounds for Jacobi heat kernel established

Explicit control of multiplicative constants achieved

Bounds extended to heat kernels on symmetric spaces

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.