On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds

TL;DR

This paper investigates the regularity of solutions to the $ar{ ext{d}}$-problem on certain complex domains with $C^2$ boundaries, establishing conditions under which solutions gain fractional derivatives depending on boundary Levi eigenvalues.

Contribution

It provides new regularity results for $ar{ ext{d}}$-solutions on $a_q$ domains with $C^2$ boundary, including fractional derivative gains under specific Levi eigenvalue conditions.

Findings

Solutions gain 1/2 derivative when $q=1$ and $f$ is in $ ext{Lambda}^r$ with $r>1$.

Regularity for $q>1$ depends on boundary smoothness or Levi eigenvalues.

Existence of solutions on the closure of the domain under certain boundary conditions.

Abstract

We study regularity of solutions $u$ to $\overline\partial u=f$ on a relatively compact $C^2$ domain $D$ in a complex manifold of dimension $n$, where $f$ is a $(0,q)$ form. Assume that there are either $(q+1)$ negative or $(n-q)$ positive Levi eigenvalues at each point of boundary $\partial D$. Under the necessary condition that a locally $L^2$ solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$ derivative when $q=1$ and $f$ is in the H\"older-Zygmund space $\Lambda^r(\overline D)$ with $r>1$. For $q>1$, the same regularity for the solutions is achieved when $\partial D$ is either sufficiently smooth or of $(n-q)$ positive Levi eigenvalues everywhere on $\partial D$.