The canonical Jacobi–Dunkl transform and its applications

In this paper, we introduce the canonical Jacobi–Dunkl transform and study some applications of this transform. We study the generalized translation operator associated with the square of the Jacobi–Dunkl operator and present some basic properties. Then, we define the generalized convolution product associated with the canonical Jacobi–Dunkl transform and establish Young's inequality. As the applications, we investigate some qualitative uncertainty principles and provide the solutions of the generalized heat and Schrödinger's equations associated with this transform. In particular, we establish Hardy's, Beurling's and Miyachi's uncertainty principles for the canonical Jacobi–Dunkl transform.