Common boundary values of holomorphic functions for two-sided complex structures

TL;DR

This paper investigates boundary behavior of functions holomorphic with respect to two different almost complex structures sharing a boundary, establishing smoothness under a specific norm condition without requiring integrability.

Contribution

It extends boundary regularity results for holomorphic functions to the setting of almost complex structures, relaxing the integrability condition.

Findings

Functions holomorphic in two structures are smooth up to the shared boundary under a norm condition.

The methods apply to almost complex structures, not just integrable complex structures.

The result generalizes classical boundary regularity theorems to a broader geometric context.

Abstract

Let $\Omega_1,\Omega_2$ be two disjoint open sets in $\mathbf C^n$ whose boundaries share a smooth real hypersurface $M$ as relatively open subsets. Assume that $\Omega_i$ is equipped with a complex structure $J^i$ which is smooth up to $M$. Assume that the operator norm $\|J^2-J^1\|<2$ on $M$. Let $f$ be a continuous function on the union of $\Omega_1,\Omega_2, M$. If $f$ is holomorphic with respect to both structures in the open sets, then $f$ must be smooth on the union of $\Omega_1$ with $M$. Although the result as stated is far more meaningful for integrable structures, our methods make it much more natural to deal with the general almost complex structures without the integrability condition. The result is therefore proved in the framework of almost complex structures.