Boundary cross theorem in dimension 1 with singularities Viet-Anh NguyenPeter Pflug 32D1532D10boundary cross theoremfiberwise polar/discreteholomorphic extensionharmonic measure Let $D$ and $G$ be copies of the open unit disc in $\mathbb{C}$, let $A$ (resp. $B$) be a measurable subset of $\:\partial D$ (resp. $\partial G$), let $W$ be the $2$-fold cross $((D\cup A)\times B)\cup(A\times(B\cup G))$, and let $M$ be a relatively closed subset of $\:W$. Suppose in addition that $A$ and $B$ are of positive one-dimensional Lebesgue measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that $M\cap(A\times B) = \varnothing$. We determine the "envelope of holomorphy" $\widehat{W\setminus M}$ of $\:W\setminus M$ in the sense that any function locally bounded on $W\setminus M$, measurable on $A\times B$, and separately holomorphic on $((A\times G)\cup(D\times B))\setminus M$ "extends" to a function holomorphic on $\widehat{W\setminus M}$. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3478 10.1512/iumj.2009.58.3478 en Indiana Univ. Math. J. 58 (2009) 393 - 414 state-of-the-art mathematics http://iumj.org/access/