# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Meromorphic extensions from small families of circles and holomorphic extensions from spheresHTML articles powered by AMS MathViewer

by Josip Globevnik
Trans. Amer. Math. Soc. 364 (2012), 5857-5880 Request permission

## Abstract:

Let $\mathbb {B}$ be the open unit ball in $\mathbb {C}^2$ and let $a, b, c$ be three points in $\mathbb {C}^2$ which do not lie in a complex line, such that the complex line through $a, b$ meets $\mathbb {B}$ and such that if one of the points $a, b$ is in $\mathbb {B}$ and the other in $\mathbb {C}^2\setminus \overline {\mathbb {B}}$ then $\langle a|b\rangle \not = 1$ and such that at least one of the numbers $\langle a|c\rangle ,\ \langle b|c\rangle$ is different from $1$. We prove that if a continuous function $f$ on $b\mathbb {B}$ extends holomorphically into $\mathbb {B}$ along each complex line which meets $\{ a, b, c\}$, then $f$ extends holomorphically through $\mathbb {B}$. 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 $\mathbb {B}$. 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 $\Delta$ be the open unit disc in $\mathbb {C}$. Given $\alpha \in \Delta$ let $\mathcal {C}_\alpha$ 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 $\alpha , \beta \in \Delta ,\ \alpha \not = \beta$, and $n\in \mathbb {N}$, a continuous function $f$ on $\overline {\Delta }$ extends meromorphically from every circle $\Gamma \in \mathcal {C}_\alpha \cup \mathcal {C}_\beta$ through the disc bounded by $\Gamma$ with the only pole at the center of $\Gamma$ of degree not exceeding $n$ if and only if $f$ is of the form $f(z) = a_0(z)+a_1(z)\overline z +\cdots +a_n(z)\overline z^n (z\in \Delta )$ where the functions $a_j, 0\leq j\leq n$, are holomorphic on $\Delta$.
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