Extremal discs and the holomorphic extension from convex hypersurfaces

Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f holomorphically extends to D. The result partially answers a conjecture by Globevnik and Stout of 1991

Holomorphic embeddings of planar domains into C 2

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46245/1/208_2005_Article_BF01461006.pd

Boundary functions on a bounded balanced domain

summary:We solve the following Dirichlet problem on the bounded balanced domain $\Omega$ with some additional properties: For $p>0$ and a positive lower semi-continuous function $u$ on $\partial \Omega$ with $u(z)=u(\lambda z)$ for $|\lambda |=1$, $z\in \partial \Omega$ we construct a holomorphic function $f\in \Bbb O(\Omega )$ such that $u(z)=\int _{\Bbb Dz}|f|^pd \frak L_{\Bbb Dz}^2$ for $z\in \partial \Omega$, where $\Bbb D=\{\lambda \in \Bbb C\:|\lambda |<1\}$