The heat semigroup associated with the Jacobi--Cherednik operator and its applications

Abstract

In this paper, we study the heat equation associated with the Jacobi--Cherednik operator on the real line. We establish some basic properties of the Jacobi--Cherednik heat kernel and heat semigroup. We also provide a solution to the Cauchy problem for the Jacobi--Cherednik heat operator and prove that the heat kernel is strictly positive. Then, we characterize the image of the space $L^2(\mathbb{R},A_{\alpha ,\beta })$ under the Jacobi--Cherednik heat semigroup as a reproducing kernel Hilbert space. As an application, we solve the modified Poisson equation and present the Jacobi--Cherednik--Markov processes.

References in corpus (2)

• Holomorphic Sobolev spaces and the generalized Segal-Bargmann transform
• Opdam's hypergeometric functions: product formula and convolution structure in dimension 1