The Neumann problem for the $k$-Cauchy-Fueter complexes over $k$-pseudoconvex domains in $\mathbb{R}^4$ and the $L^2$ estimate

TL;DR

This paper develops an $L^2$ estimate and solves the Neumann problem for $k$-Cauchy-Fueter complexes on $k$-pseudoconvex domains in $\

Contribution

It introduces $k$-pseudoconvex domains and establishes the $L^2$ estimate for the Neumann problem in quaternionic analysis, extending complex analysis techniques to quaternionic variables.

Findings

Established $L^2$ estimate for the Neumann problem

Solved the Neumann problem over $k$-pseudoconvex domains

Proved a vanishing theorem for first cohomology groups

Abstract

The $k$-Cauchy-Fueter operators and complexes are quaternionic counterparts of the Cauchy-Riemann operator and the Dolbeault complex in the theory of several complex variables. To develop the function theory of several quaternionic variables, we need to solve the non-homogeneous $k$-Cauchy-Fueter equation over a domain under the compatibility condition, which naturally leads to a Neumann problem. The method of solving the $\overline{\partial}$-Neumann problem in the theory of several complex variables is applied to this Neumann problem. We introduce notions of $k$-plurisubharmonic functions and $k$-pseudoconvex domains, establish the $L^2$ estimate and solve this Neumann problem over $k$-pseudoconvex domains in $\mathbb{R}^4$. Namely, we get a vanishing theorem for the first cohomology groups of the $k$-Cauchy-Fueter complex over such domains.