Testing families of analytic discs

It is a well-known fact in the theory of several complex variables that a function
is holomorphic if and only if it is holomorphic in each variable separately. This
result goes back to Hartogs. It is natural to consider a boundary version of Hartogs’ theorem. The general problem is to take a boundary function and ask if holomorphic extensions on some families of complex curves are enough to guarantee an extension which is holomorphic in all variables simultaneously.
We will talk about the known results on the subject and show some new results obtained in collaboration with M. Fassina and S. Pinton for the spiecial case of the unit ball in C^n.