Not quite sure if this is what you're looking for, but the following paper by Beals, Gaveau and Greiner used what is essentially the method of stationary phase to get uniform upper bounds on the subelliptic heat kernel for Heisenberg groups. In this case an exact formula for the heat kernel was already known (as, essentially, the Fourier transform of the Mehler kernel), but since it involves an oscillatory integral, it is not so easy to read off bounds from the formula.

Many other authors have used similar techniques to get either bounds or asymptotics for heat kernels of various kinds. There is a paper of Gaveau from 1977, studying asymptotics for the subelliptic heat kernel on the 3-dimensional Heisenberg group, that I think sort of kicked it off.

Beals, Richard; Gaveau, Bernard; Greiner, Peter C.: Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl. (9) 79 (2000), no. 7, 633–689. MR 1776501, DOI 10.1016/S0021-7824(00)00169-0