Boundary Traces of Holomorphic Functions on the Unit Ball in Cⁿ

Abstract It is a classical theorem that if a function on the unit circle is integrable, then it is the nontangential limit of a holomorphic function on the open disc (subject to a certain growth condition) 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 a “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.