Meromorphic extensions from small families of circles and holomorphic extensions from spheres

Abstract

Let B be the open unit ball in C^2 and let a, b, c be three points in C^2 which do not lie in a complex line, such that the complex line through a and b meets B and such that is different from 1 if one of the points a, b is in B and the other in the complement of B and such that at least one of the numbers , is different from 1. We prove that if a continuous function f on the sphere bB extends holomorphically into B along each complex line which passes through one of the points a, b, c then f extends holomorphically through B. This generalizes recent work of L.Baracco who proved such a result in the case when the points a, b, c are contained in B. The proof is different from the one of Baracco and uses the following one variable result which we also prove in the paper and which in the real analytic case follows from the work of M.Agranovsky: Let D be the open unit disc in C. Given a in D let C(a) be the family of all circles in D obtained as the images of circles centered at the origin under an automorphism of D that maps the origin to a. Given distinct points a, b in D and a positive integer n, a continuous function f on the closed unit disc extends meromorphically from every circle T in either C(a) or C(b) through the disc bounded by T with the only pole at the center of T of degree not exceeding n if and only if f is of the form f(z) = g_0(z)+g_1(z)\bar z +...+ g_n(z)\bar z^n where the functions g_0, g_1, ..., g_n are holomorphic on D.