We consider holomorphic functions on a non-singular subvariety of a smoothly bounded convex domain of finite type. A sufficient and necessary condition is proved for such a function to have an extension to a $p$-integrable holomorphic function on the whole domain. This is shown under transversallity assumption and certain non-degeneracy condition of the subvariety.
"Extension and restriction of holomorphic functions on convex finite type domains." Illinois J. Math. 54 (2) 509 - 542, Summer 2010. https://doi.org/10.1215/ijm/1318598671