# Hausdorff-$(2n-2)$ dimensional measure zero set and compactness of the $\overline{\partial}$-Neumann operator on $(0,n-1)$ forms

## TL;DR

This paper establishes a connection between the measure-zero property of weakly pseudoconvex boundary points and the compactness of the $ar{
abla}$-Neumann operator on certain forms, using a variant of Catlin's Property (P_q).

## Contribution

It demonstrates that if the Hausdorff $(2n-2)$-dimensional measure of weakly pseudoconvex boundary points is zero, then the $ar{
abla}$-Neumann operator is compact on $(0,n-1)$ forms, extending previous results.

## Key findings

- Hausdorff measure zero implies compactness of the $ar{
abla}$-Neumann operator.
- Uses a variant of Property (P_q) to relate boundary geometry to operator compactness.
- Provides conditions under which the $ar{
abla}$-Neumann operator is compact on $(0,n-1)$ forms.

## Abstract

By using a variant Property $(P_q)$ of Catlin, we discuss the relation of small set of weakly pseudoconvex points on the boundary of pseudoconvex domain and compactness of the $\overline{\partial}$-Neumann operator. In particular, we show that if the Hausdorff $(2n-2)$-dimensional measure of the weakly pseudoconvex points on the boundary of a smooth bounded pseudoconvex domain is zero, then the $\overline{\partial}$-Neumann operator $N_{n-1}$ is compact on $(0,n-1)$-level $L^2$-integrable forms.