On the Bergman kernel in weighted monogenic Bargmann-Fock spaces

TL;DR

This paper investigates the Bergman kernel in weighted monogenic Bargmann-Fock spaces within Clifford algebra, establishing estimates and existence results using $L^2$-methods and subharmonic inequalities.

Contribution

It introduces new $L^2$-estimates and subharmonic inequalities for Clifford algebra, proving the existence and providing estimates of the Bergman kernel in this setting.

Findings

Existence of the Bergman kernel $B_(x,y)$ in weighted monogenic Bargmann-Fock spaces.

Derived estimates of $B_(x,y)$ on and off the diagonal.

Obtained an upper bound for the weighted harmonic Bergman kernel.

Abstract

In this paper, we study the Bergman kernel $B_\varphi(x,y)$ of generalized Bargmann-Fock spaces in the setting of Clifford algebra. The versions of $L^2$-estimate method and weighted subharmonic inequality for Clifford algebra are established. Consequently we show the existence of $B_\varphi(x,y)$ and then give some estimates on and off the diagonal. As a by-product, we also obtain an upper estimate of the weighted harmonic Bergman kernel.