# Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

## TL;DR

This paper establishes precise two-sided heat kernel estimates on compact rank-one symmetric spaces, Jacobi expansions, balls, and simplices, improving upon previous Gaussian bounds with more accurate results.

## Contribution

The authors extend sharp heat kernel estimates to a broader class of spaces and expansions, generalizing prior results on Euclidean spheres and providing more precise bounds.

## Key findings

- Sharp two-sided heat kernel estimates on compact rank-one symmetric spaces
- Precise bounds for Jacobi expansions on a ball and a simplex
- Improved accuracy over previous Gaussian estimates

## Abstract

We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat kernel bounds are shown in the context of classical Jacobi expansions, on a ball and on a simplex. These results are more precise than the qualitatively sharp Gaussian estimates proved recently by several authors.