Sums of CR and projective dual CR functions

TL;DR

This paper characterizes when a smooth function on a strongly complex convex hypersurface in complex projective space can be expressed as a sum of a CR function and a dual CR function, using differential forms and tangential vector fields.

Contribution

It provides new characterizations for sums of CR and dual CR functions on complex hypersurfaces, extending previous work with differential forms and vector fields.

Findings

Characterization via differential forms

Characterization using tangential vector fields

Extension of Lee's pluriharmonic boundary value work

Abstract

A smooth, strongly $\mathbb{C}$-convex, real hypersurface $S$ in $\mathbb{CP}^n$ admits a projective dual CR structure in addition to the standard CR structure. Given a smooth function $u$ on $S$, we provide characterizations for when $u$ can be decomposed as a sum of a CR function and a dual CR function. Following work of Lee on pluriharmonic boundary values, we provide a characterization using differential forms. We further provide a characterization using tangential vector fields in the style of Audibert and Bedford.