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

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 𝐿2⁡(ℝ,𝐴𝛼,β) 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.