Heat kernels of the discrete Laguerre operators

TL;DR

This paper explicitly computes heat kernels for discrete Laguerre operators using Jacobi polynomials, demonstrating ultracontractivity, and establishing key properties and inequalities related to the associated Markov semigroup.

Contribution

It provides explicit formulas for heat kernels of discrete Laguerre operators and extends classical inequalities to their perturbations.

Findings

Heat kernels expressed via Jacobi polynomials

Heat semigroup shown to be ultracontractive

Established Hardy inequality and perturbation estimates

Abstract

For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel-Lieb-Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.