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.Comment: 16 page