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

Abstract

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.

Application of the hypercomplex fractional integro-differential operators to the fractional Stokes equation

“…The previous definition allows us to rewrite (11) in the alternative form T αRL D α a + g (x) + (F α g) (x) = g(x), with x ∈ Ω. For the regularity and mapping properties of (12) we refer to [2]. Again, in [2] we proved the following result:…”

“…xi are the left Riemann-Liouville and Caputo fractional derivatives (2) and (4) of orders 1 + α i and 1+αi 2 , with respect to the variable [2]). Due to the nature of the eigenfunctions and the fundamental solution of these operators we additionally need to consider the variable…”

“…Next recalling from [2] we know that a family of fundamental solutions of the fractional Dirac operator C D α a + can be represented in the way C G α…”

“…In this section we recall the definitions and the main properties of the fractional versions of the Teodorescu and Cauchy-Bitsadze operators that are going to be used in the sequel to treat the fractional Stokes problem. For all the detailed proofs and calculations we refer to our paper [2]. First we recall the following fractional Stokes formula…”

“…Replacing f by C G α + (x−y) in (10) we now may obtain the following fractional Borel-Pompeiu formula (a detailed proof is presented in [2]). Theorem 2.2 Let g ∈ C 0,n (Ω) ∩ AC 1 (Ω) ∩ AC(Ω).…”