Sharp estimates for Jacobi heat kernels in double conic domains

TL;DR

This paper derives sharp estimates for Jacobi heat kernels in double conic, hyperboloid, and paraboloid domains, extending previous methods to new geometric settings and discussing the scope of these estimates.

Contribution

It provides the first sharp heat kernel estimates for Jacobi kernels in double conic and hyperbolic domains, integrating polynomial frameworks with modern analytical techniques.

Findings

Established sharp estimates for Jacobi heat kernels in double conic domains.

Extended the framework of heat kernel estimates to hyperbolic and paraboloid settings.

Identified limitations in applying these estimates to certain geometric configurations.

Abstract

We study the even and odd Jacobi heat kernels defined in the context of the multidimensional double cone and its surface, the multidimensional hyperboloid and its surface, and the multidimensional paraboloid and its surface. By integrating the framework of Jacobi polynomials on these domains, as analyzed by Xu, with contemporary methods developed by Nowak, Sj\"ogren, and Szarek for obtaining sharp estimates of the spherical heat kernel, we establish genuinely sharp estimates for the even and odd Jacobi heat kernel in the double conic settings, and for the even Jacobi heat kernel on hyperbolic settings. We also discuss the limitations encountered in extending these results to the remaining settings.