On holomorphic continuation of integrable functions along finite families of complex lines in an $n$-circular domain

This paper contains some results related to holomorphic extension of integrable functions defined on the boundary of $D\subset\mathbb C^n$, $n>1$ into this domain. We shall consider integrable functions with the property of holomorphic extension along complex lines. In the complex plane $\mathbb C$ the results about functions with such property are trivial. Therefore, our results are essentially multidimensional.