Localized versions of function spaces and generic results

Abstract

We consider generalizations of classical function spaces by requiring that a holomorphic in $\Omega$ function satisfies some property when we approach from $\Omega$, not the whole boundary, but only a part of it. These spaces endowed with their natural topology are Fr$\acute{e}$chet spaces. We prove some generic non-extendability results in such spaces and generic nowhere differentiability on the corresponding part of $\partial \Omega$.

Section 9 has been deleted, fixed typos, revised format, 15 pages