Sharp estimates for Jacobi heat kernels in conic domains

TL;DR

This paper derives precise estimates for Jacobi heat kernels on conic domains, advancing understanding of heat propagation in these geometries by combining polynomial theory and recent sharp estimate techniques.

Contribution

It introduces sharp heat kernel estimates on conic domains by integrating Jacobi polynomial theory with novel estimation methods.

Findings

Established sharp bounds for Jacobi heat kernels in conic geometries.

Extended techniques from spherical heat kernel estimates to conic domains.

Enhanced mathematical understanding of heat diffusion in multidimensional cones.

Abstract

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 $\mathbb{V}^{d+1}_0$. To do so, we combine the theory of Jacobi polynomials on the cone explored by Xu with the recent techniques by Nowak, Sj\"ogren, and Szarek, developed to find genuinely sharp estimates for the spherical heat kernel.