Ferreira, M.Kraußhar, R. S.Rodrigues, M. M.Vieira, N.2021-03-192021-03-192019-02-24Ferreira, M, Krauβhar, RS, Rodrigues, MM, Vieira, N. A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus. Math Meth Appl Sci. 2019; 42(10): 3633– 3653. https://doi.org/10.1002/mma.56021099-14760170-4214http://hdl.handle.net/10400.8/5525Acknowledgements: The work of M. Ferreira, M.M. Rodrigues and N. Vieira was supported by Portuguese funds through CIDMACenter for Research and Development in Mathematics and Applications, and FCT–Fundação para a Ciência e a Tecnologia, within project UID/MAT/04106/2019. The work of the authors was supported by the project New Function Theoretic Methods in Computational Electrodynamics / Neue funktionentheoretische Methoden f¨ur instation¨are PDE, funded by Programme for Cooperation in Science between Portugal and Germany (“Programa de A¸c˜oes Integradas Luso-Alem˜as/2017” - Acção No. A-19/08 - DAAD-PPP Deutschland-Portugal, Ref: 57340281). N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).In this paper, we develop a fractional integro-differential operator calculus for Clifford-algebra valued functions. To do that we introduce fractional analogs 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.engFractional Clifford analysisFractional derivativesStokes's formulaBorel-Pompeiu formulaCauchy's integral formulaHodge-type decompositionA higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculusjournal articlehttps://doi.org/10.1002/mma.5602