New results from old investigation: a note on fractional \(m\)-dimensional differential operators. The fractional Laplacian (Q2017457)

The aim of this paper is to highlight that in his famous works, Riesz gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian, which can be applied to much wider domains of functions than those given in the current literature and are based in the theory of fractional power of operators or in certain hyper-singular integrals. The authors show that the \(m\)-dimensional integral operators given by Riesz, their properties and the well-known techniques used to give the wider definition for many of the known one-dimensional fractional differential operators allow to get a more suitable explicit definition of the differential fractional \(m\)-dimensional coupled Laplacian. Furthermore, in the last part of the paper, the authors introduce the corresponding fractional hyperbolic differential operator, called the fractional Lorentzian Laplacian.