The Canonical Opdam–Cherednik Transform and Its Applications

In this paper, we introduce the canonical Opdam–Cherednik transform and study some of its important properties. We define the generalized translation operator associated with the square of the Jacobi–Cherednik operator and present some basic properties. We also study the generalized convolution product associated with the canonical Opdam–Cherednik transform and establish Young’s inequality. Nonetheless, special attention is paid to the applications of this new transform. At the heart of our investigation lies Hardy’s, Beurling’s and Miyachi’s uncertainty principles, and some applications on the generalized heat and Schrödinger’s equations.