On the fractional Laplace-Bessel operator

Abstract

In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator $ \Delta_{\nu} $, defined as$ \Delta_{\nu} = \sum\limits_{i = 1}^{n}\frac{\partial^2}{\partial x_{i}^2} + \frac{\nu_i}{x_{i}}\frac{\partial}{\partial x_{i}}, \quad \nu_i\geq 0. $The fractional power of this operator is introduced as a pseudo-differential operator through the multi-dimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations.

Keywords

• laplace-bessel operator
• fractional calculus
• bessel function