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

Abstract

Let B \mathbb {B} be the open unit ball in C 2 \mathbb {C}^2 and let a , b , c a, b, c be three points in C 2 \mathbb {C}^2 which do not lie in a complex line, such that the complex line through a , b a, b meets B \mathbb {B} and such that if one of the points a , b a, b is in B \mathbb {B} and the other in C 2 ∖ B ¯ \mathbb {C}^2\setminus \overline {\mathbb {B}} then ⟨ a | b ⟩ ≠ 1 \langle a|b\rangle \not = 1 and such that at least one of the numbers ⟨ a | c ⟩ , ⟨ b | c ⟩ \langle a|c\rangle ,\ \langle b|c\rangle is different from 1 1 . We prove that if a continuous function f f on b B b\mathbb {B} extends holomorphically into B \mathbb {B} along each complex line which meets { a , b , c } \{ a, b, c\} , then f f extends holomorphically through B \mathbb {B} . This generalizes the recent result of L. Baracco who proved such a result in the case when the points a , b , c a, b, c are contained in B \mathbb {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 Δ \Delta be the open unit disc in C \mathbb {C} . Given α ∈ Δ \alpha \in \Delta let C α \mathcal {C}_\alpha be the family of all circles in Δ \Delta obtained as the images of circles centered at the origin under an automorphism of Δ \Delta that maps 0 0 to α \alpha . Given α , β ∈ Δ , α ≠ β \alpha , \beta \in \Delta ,\ \alpha \not = \beta , and n ∈ N n\in \mathbb {N} , a continuous function f f on Δ ¯ \overline {\Delta } extends meromorphically from every circle Γ ∈ C α ∪ C β \Gamma \in \mathcal {C}_\alpha \cup \mathcal {C}_\beta through the disc bounded by Γ \Gamma with the only pole at the center of Γ \Gamma of degree not exceeding n n if and only if f f is of the form f ( z ) = a 0 ( z ) + a 1 ( z ) z ¯ + ⋯ + a n ( z ) z ¯ n ( z ∈ Δ ) f(z) = a_0(z)+a_1(z)\overline z +\cdots +a_n(z)\overline z^n (z\in \Delta ) where the functions a j , 0 ≤ j ≤ n a_j, 0\leq j\leq n , are holomorphic on Δ \Delta .