Partially holomorphic functions in several variables

In this talk we introduce the notion of “partially holomorphic functions” in several variables. Given a domain in $\mathbb{C}^n$, we all a function partially holomorphic if it is holomorphic on the slices in one affine direction, while in one or more other affine directions it satisfies moment conditions for being holomorphic. On the ball an example is any polynomial in $z$, $w$, $|w|$. Any such function is holomorphic in $z$, while on slices where $z$ is constant, it extends holomorphically from any circle centered at 0. Partially holomorphic functions on the ball are the main object of study in this talk. They arise naturally when considering functions on the sphere which have holomorphic extensions in multiple affine directions. Other domains are also interesting. The goal of this research was to show that partially holomorphic functions on the ball must be real analytic, given enough directions; we have partial progress on this question.