Sharp estimates of the Jacobi heat kernel

TL;DR

This paper derives precise estimates for the Jacobi heat kernel and Poisson-Jacobi kernel, enabling analysis of their behavior and establishing weak type inequalities for the associated heat semigroup.

Contribution

It provides the first sharp magnitude estimates for the Jacobi heat kernel and Poisson-Jacobi kernel, which were previously not explicitly known.

Findings

Established sharp bounds for the Jacobi heat kernel.

Proved weak type (1,1) inequality for the Jacobi heat semigroup.

Derived sharp estimates for the Poisson-Jacobi kernel.

Abstract

The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the two other classical systems of orthogonal polynomials. We deduce sharp estimates giving the order of magnitude of this kernel, for type parameters $\alpha,\beta \ge -1/2$. As an application of the upper bound obtained, we show that the maximal operator of the multi-dimensional Jacobi heat semigroup satisfies a weak type $(1,1)$ inequality. We also obtain sharp estimates of the Poisson-Jacobi kernel.