On compactness and $L^p$-regularity in the $\overline{\partial}$-Neumann problem

TL;DR

This paper proves that compactness of the $ar{ abla}$-Neumann operator on certain domains implies $L^p$-regularity of the embedding operator for all $p$ between 2 and infinity, in the context of complex analysis.

Contribution

It establishes a link between the compactness of the $ar{ abla}$-Neumann operator and $L^p$-regularity of the embedding operator in pseudoconvex domains.

Findings

Compactness of $N_1$ implies $L^p$-regularity for all $p< abla$

Results hold for $C^4$-smooth bounded pseudoconvex domains in $C^2$

Enhances understanding of regularity properties in the $ar{ abla}$-Neumann problem

Abstract

Let $\Omega$ be a $C^4$-smooth bounded pseudoconvex domain in $\mathbb{C}^2$. We show that if the $\overline{\partial}$-Neumann operator $N_1$ is compact on $L^2_{(0,1)}(\Omega)$ then the embedding operator $\mathcal{J}:Dom(\overline{\partial})\cap Dom(\overline{\partial}^*) \to L^2_{(0,1)}(\Omega)$ is $L^p$-regular for all $2\leq p<\infty$.