  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.

Complex Variables