Sharp estimates for Jacobi heat kernels in conic domains

We prove genuinely sharp estimates for the Jacobi heat kernels introduced in the context of the multidimensional cone ${\mathbb{V}}^{d+1}$ and its surface ${\mathbb{V}}_0^{d+1}$. To do so, we combine the theory of Jacobi polynomials on the cone explored by Xu with the recent techniques by Nowak, Sjögren, and Szarek, developed to find genuinely sharp estimates for the spherical heat kernel.