# Schatten class Hankel and $\overline{\partial}$-Neumann operators on pseudoconvex domains in $\mathbb{C}^n$

## TL;DR

This paper investigates the Schatten class properties of Hankel and $ar{
abla}$-Neumann operators on smooth pseudoconvex domains in complex space, establishing conditions under which these operators are compact or belong to specific Schatten classes.

## Contribution

It proves that certain Hankel operators are in Schatten $p$-class only for constant symbols when $p o 2n$, and demonstrates the $ar{
abla}$-Neumann operator is not Hilbert-Schmidt on these domains.

## Key findings

- Hankel operators in Schatten $p$-class imply constant symbols for $p o 2n$.
- The $ar{
abla}$-Neumann operator is not Hilbert-Schmidt.
- Provides conditions linking operator class membership to function constancy.

## Abstract

Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt.