Integrating holomorphic $L^1$-functions

TL;DR

This paper demonstrates that integrals of holomorphic functions in $L^1$ can be represented via smooth functions with boundary vanishing properties, with applications to the Bergman projection's smoothing effects.

Contribution

It introduces a novel representation of $L^1$ holomorphic functions using boundary-vanishing smooth functions and explores implications for the Bergman projection.

Findings

Integral representation using boundary-vanishing smooth functions

Smoothing property of the Bergman projection for conjugate holomorphic functions

Extension of integral representation techniques to complex domains

Abstract

Let $\Omega\Subset\mathbb{C}^{n}$ be a domain with smooth boundary, $k\in\mathbb{N}$. It is shown that the integral of a holomorphic function in $L^1(\Omega)$ may be represented as the integral of this function against a smooth function vanishing to order $k-1$ on $b\Omega$. An application for a smoothing property of the Bergman projection for conjugate holomorphic functions is given.