Boundary Traces of Holomorphic Functions on the Unit Ball in $\mathbb{C}^n$

Abstract

It is a classical theorem that if a function is integrable along the boundary of the unit circle, then the function is the nontangential limit of a holomorphic function on the open disc if and only if its Fourier coefficients for nonnegative integers are zero. In this article we generalize this result to higher complex dimensions by proving that for an integrable function on the unit sphere, it is ``boundary trace'' of a holomorphic function on the open unit ball if and only if two particular families of integral equations are satisfied. To do this, we use the theory of Hardy spaces as well as the invariant Poisson and Cauchy integrals. This article is written in a style that is meant to be welcoming to those who have taken a course in complex analysis but who are not necessarily experts in the field.