The tangential $k$-Cauchy-Fueter complexes and Hartogs' phenomenon over the right quaternionic Heisenberg group

TL;DR

This paper develops a quaternionic analogue of the $ar{oundary}_b$-complex on the Heisenberg group, constructs related complexes, and proves Hartogs' extension phenomenon for quaternionic CR functions using $L^2$ estimates.

Contribution

It introduces the tangential $k$-Cauchy-Fueter complexes on the quaternionic Heisenberg group and establishes Hartogs' phenomenon for $k$-CF functions.

Findings

Construction of tangential $k$-Cauchy-Fueter complexes.

Solution of the nonhomogeneous equation via $L^2$ estimates.

Proof of Hartogs' extension phenomenon for $k$-CF functions.

Abstract

We construct the tangential $k$-Cauchy-Fueter complexes on the right quaternionic Heisenberg group, as the quaternionic counterpart of $\overline{\partial}_b$-complex on the Heisenberg group in the theory of several complex variables. We can use the $L^2$ estimate to solve the nonhomogeneous tangential $k$-Cauchy-Fueter equation under the compatibility condition over this group modulo a lattice. This solution has an important vanishing property when the group is higher dimensional. It allows us to prove the Hartogs' extension phenomenon for $k$-CF functions, which are the quaternionic counterpart of CR functions.