Necessary conditions for H\"older regularity gain of d-bar equation in C^3

TL;DR

This paper establishes an upper bound on the H"older regularity gain for solutions of the ar-equation in ^3, based on the order of contact of a holomorphic curve at the boundary of a pseudoconvex domain.

Contribution

It provides a necessary condition linking the order of contact of a holomorphic curve to the maximal Hf6lder regularity gain for ar-solutions in ^3.

Findings

Maximal Hf6lder gain is at most 1/ta.

The result relates geometric boundary properties to regularity of solutions.

Provides a geometric criterion for regularity improvement limits.

Abstract

Suppose that a smooth holomorphic curve $V$ has order of contact $\eta$ at a point $w_0$ in the boundary of a pseudoconvex domain $\Omega$ in $\mathbb{C}^3.$ We show that the maximal gain in H\"older regularity for solutions of the $\bar{\partial}$-equation is at most $\frac{1}{\eta}.$