The Neumann Problem for the k-Cauchy–Fueter Complex over k-Pseudoconvex Domains in $$\mathbb {R}^4$$ R 4 and the $$L^2$$ L 2 Estimate

Abstract

The k-Cauchy–Fueter operator and complex are quaternionic counterparts of the Cauchy–Riemann operator and the Dolbeault complex in the theory of several complex variables, respectively. 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 the Neumann problem over k-pseudoconvex domains in $$\mathbb {R}^4$$ . Namely, we get a vanishing theorem for the first cohomology group of the k-Cauchy–Fueter complex over such domains.