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

## TL;DR

This paper develops weighted $L^2$ estimates for the $k$-Cauchy-Fueter operator, introduces the weighted $k$-Bergman space and projection, and analyzes the asymptotic behavior of the associated kernel function, extending complex analysis concepts to quaternionic settings.

## Contribution

It establishes weighted $L^2$ estimates for the $k$-Cauchy-Fueter operator and introduces the $k$-Bergman projection and kernel, providing new tools for quaternionic analysis.

## Key findings

- Weighted $L^2$ estimates for the $k$-Cauchy-Fueter operator.
- Introduction of the $k$-Bergman orthogonal projection and kernel.
- Asymptotic decay properties of the kernel function.

## 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.