Analyticity on translates of a Jordan curve

Abstract:

Let $\Omega$ be a domain in $\mathbb {C}$ which is symmetric with respect to the real axis and whose boundary is a real analytic simple closed curve. Translate $\overline \Omega$ vertically to get $K = \bigcup \{ \overline \Omega +it,\ -r\leq t\leq r\}$ where $r>0$ is such that $(\overline \Omega -ir)\cap (\overline \Omega +ir)= \emptyset$. We prove that if $f$ is a continuous function on $K$ such that for each $t,\ -r\leq t\leq r$, the function $f|(b\Omega +it)$ has a continuous extension to $\overline \Omega +it$ which is holomorphic on $\Omega +it$, then $f$ is holomorphic on $\textrm {Int}K$.