A higher dimensional fractional Borel-Pompeiu formula and a related hypercomplex fractional operator calculus

摘要

In this paper, we develop a fractional integro-differential operator calculus for Clifford algebra-valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy-Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel-Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge-type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann-Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.