On the weighted $L^2$ estimate for the $k$-Cauchy-Fueter operator and the weighted $k$-Bergman kernel

Abstract

The $k$-Cauchy-Fueter operators, $k=0,1,\ldots$, are quaternionic counterparts of the Cauchy-Riemann operator in the theory of several complex variables. The weighted $L^2$ method to solve Cauchy-Riemann equation is applied to find the canonical solution to the non-homogeneous $k$-Cauchy-Fueter equation in a weighted $L^2$-space, by establishing the weighted $L^2$ estimate. The weighted $k$-Bergman space is the space of weighted $L^2$ integrable functions annihilated by the $k$-Cauchy-Fueter operator, as the counterpart of the Fock space of weighted $L^2$-holomorphic functions on $\mathbb{C}^n$. 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.