Orthogonal testing families and holomorphic extension from the sphere to the ball

Let $\mathbb{B}^2$ denote the open unit ball in $\mathbb{C}^2$, and let $p\in \mathbb{C}^2\setminus\overline{\mathbb{B}^2}$. We prove that if $f$ is an analytic function on the sphere $\partial\mathbb{B}^2$ that extends holomorphically in each variable separately and along each complex line through $p$, then $f$ is the trace of a holomorphic function in the ball.Comment: 9 pages, 2 figures. Final version to appear in Math.

Holomorphic extension from the sphere to the ball

Real analytic functions on the boundary of the sphere which have separate holomorphic extension along the complex lines through a boundary point have holomorphic extension to the ball. This was proved in a previous preprint by an argument of CR geometry. We give here an elementary proof based on the expansion in holomorphic and antiholomorphic powers

On the Ekeland-Hofer symplectic capacities of the real bidisc

In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2\times D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the first, second and third Ekeland-Hofer symplectic capacity of $D^2\times D^2$. We discuss some applications to questions of symplectic rigidity.Comment: v3: Final version, to appear in "Pacific J. Math.", 20 page