The [formula omitted]-Cauchy–Fueter complex, Penrose transformation and Hartogs phenomenon for quaternionic [formula omitted]-regular functions

Abstract

By using complex geometric method associated to the Penrose transformation, we give a complete derivation of an exact sequence over C 4 n , whose associated differential complex over H n is the k -Cauchy–Fueter complex with the first operator D 0 ( k ) annihilating k -regular functions. D 0 ( 1 ) is the usual Cauchy–Fueter operator and 1 -regular functions are quaternionic regular functions. We also show that the k -Cauchy–Fueter complex is elliptic. By using the fundamental solutions to the Laplacian operators of 4 -order associated to the k -Cauchy–Fueter complex, we can establish the corresponding Bochner–Martinelli integral representation formula, solve the non-homogeneous k -Cauchy–Fueter equations and prove the Hartogs extension phenomenon for k -regular functions in any bounded domain.