The <italic>k</italic>-Cauchy-Fueter operators in several quaternionic variables

Abstract

This is a survey of the development of k -Cauchy-Fueter operators. We discuss the k -Cauchy-Fueter complexes, non-homogeneous k -Cauchy-Fueter equations, the Hartogs extension phenomenon for k -regular functions, Bochner-Martinelli integral representation formula, Penrose integral transformation and series expansion of k -regular functions, quaternionic Hardy space and Cauchy-Szego kernel, the relation between 0-Cauchy-Fueter operator and quaternionic Monge-Ampere operator, quaternionic closed positive currents and their Lelong number, Lelong-Jessen type formula, and tangential k -Cauchy-Fueter operators and complexes.