On the weighted L2 estimate for the k-Cauchy–Fueter operator and the weighted k-Bergman kernel

Abstract

The k-Cauchy–Fueter operators, k=0,1,…, are quaternionic counterparts of the Cauchy–Riemann operator in the theory of several complex variables. The weighted L2 method to solve Cauchy–Riemann equation is applied to find the canonical solution to the non-homogeneous k-Cauchy–Fueter equation in a weighted L2-space, by establishing the weighted L2 estimate. The weighted k-Bergman space is the space of weighted L2 integrable functions annihilated by the k-Cauchy–Fueter operator, as the counterpart of the Fock space of weighted L2-holomorphic functions on Cn. We introduce the k-Bergman orthogonal projection to this closed subspace, which can be nicely expressed in terms of the canonical solution operator, and its matrix kernel function. We also find the asymptotic decay for this matrix kernel function.