MEROMORPHIC EXTENSIONS FROM SMALL FAMILIES OF CIRCLES AND HOLOMORPHIC EXTENSIONS FROM SPHERES

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, b meets 1 and such that if one of the points a, b is in B and the other in C-2 \ (B) over bar then < a\b > not equal 1 and such that at least one of the numbers < a\c >, < b\c > is different from 1. We prove that if a continuous function f on bB extends holomorphically into B along each complex line which meets {a, b, c}, then f extends holomorphically through II. This generalizes the recent result of L. Baracco who proved such a result in the case when the points a, b, c are contained in B. The proof is quite different from the one of Baracco and uses the following one-variable result, which we also prove in the paper: Let Delta be the open unit disc in C. Given alpha is an element of Delta let C-alpha be the family of all circles in Delta obtained as the images of circles centered at the origin under an automorphism of Delta that maps 0 to alpha. Given alpha, beta is an element of Delta, alpha not equal beta, and n is an element of N, a continuous function f on (Delta) over bar extends meromorphically from every circle Gamma is an element of C-alpha boolean OR C-beta through the disc bounded by Gamma with the only pole at the center of Gamma of degree not exceeding a if and only if f is of the form f(z) = a(0)(z)+a(1)(z)(z) over bar + ... + a(n)(z)(z) over bar (n)(z is an element of Delta) where the functions a(j), 0 <= j <= n, are holomorphic on Delta.