Hartogs Phenomena for Discrete k -Cauchy–Fueter Operators

counterexample is constructed in this article to show the failure of Hartogs phenomena for discrete k -Cauchy–Fueter operators. The underlying reason for this fact is that the discrete uniqueness theorem no longer holds. Indeed, a discrete k -regular function with compact support may not vanish on the connected component of infinite. However, we can show that the Hartogs theorem remains valid for the pair K⊂Ω if K is convex and the distance between K and the complement set of Ω is larger than 4. To this end, we provide explicitly the discrete k -Cauchy–Fueter complex and solve the non-homogeneous discrete k -Cauchy–Fueter equations with compact support. We also show that the regular extension can be realized explicitly via the discrete Bochner–Martinelli formula.