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.
Nowak et al. (Thu,) studied this question.