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Hartogs phenomena for discrete \(k\)-Cauchy-Fueter operators. (English) Zbl 07707569

Summary: A 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 \(K\subset \Omega\) if \(K\) is convex and the distance between \(K\) and the complement set of \(\Omega\) 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.

MSC:

39A12 Discrete version of topics in analysis
39A70 Difference operators
30G35 Functions of hypercomplex variables and generalized variables
58J10 Differential complexes
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