The kk-Cauchy–Fueter complex, Penrose transformation and Hartogs phenomenon for quaternionic kk-regular functions

By using complex geometric method associated to the Penrose transformation, we give a complete derivation of an exact sequence over C4nC4n, whose associated differential complex over HnHn is the kk-Cauchy–Fueter complex with the first operator D0(k) annihilating kk-regular functions. D0(1) is the usual Cauchy–Fueter operator and 11-regular functions are quaternionic regular functions. We also show that the kk-Cauchy–Fueter complex is elliptic. By using the fundamental solutions to the Laplacian operators of 44-order associated to the kk-Cauchy–Fueter complex, we can establish the corresponding Bochner–Martinelli integral representation formula, solve the non-homogeneous kk-Cauchy–Fueter equations and prove the Hartogs extension phenomenon for kk-regular functions in any bounded domain.