Localized versions of function spaces and generic results

Abstract

We consider generalizations of classical function spaces by requiring that a holomorphic in Ω function satisfies some property when we approach from Ω, not the whole boundary ∂Ω, but only a part of it. These spaces endowed with their natural topology are Fréchet spaces. We prove some generic non-extendability results in such spaces and generic nowhere differentiability on the corresponding part of ∂Ω. AMS classification number: 30H05, 30H20, 30H50 Key words and phrases: Bounded holomorphic functions, Bergman spaces, Mergelyan's theorem, Baire category theorem, non-extendable functions, nowhere differentiable functionsIf Ω is a Jordan domain and J ⊂ ∂Ω is relatively open, we consider the space A 0 (Ω, J) to contain all holomorphic in Ω functions extending continuously on Ω ∪ J, endowed with its natural topology, see also [6]. We show that the generic function in A 0 (Ω, J) is nowhere differentiable on J. Here, the differentiability is meant with respect to the parametrization induced by any Riemann map from the open unit disc onto Ω [9], or with respect to the position [7]. We notice that in this case, polynomials are dense in A 0 (Ω, J). Furthermore, we generalize the previous results to domains Ω bounded by a finite number of disjoint Jordan curves. We also consider the spaces A p (Ω, J) containing all functions f ∈ A 0 (Ω, J), such that all the derivatives f (l) , 0 ≤ l ≤ p belong to A 0 (Ω, J), endowed with its natural topology. We show that if Ω is convex, then for the generic function f ∈ A p (Ω, J), the derivative f (p) is nowhere differentiable on J.

Generic non-extendability and total unboundedness in function spaces

“…see [4]. If V ⊂ Ω and V is bounded, then obviously X(Ω, V ) = H(Ω) and the space is endowed with its usual Fréchét topology.…”

“…a sequence of compact subsets of Ω ∪ J, see [4]. We define the space A p (Ω, J) similarly to the space A 0 (Ω, J) = A(Ω, J).…”

“…Let p ∈ (0, ∞), Ω ⊂ C a domain and V ⊂ Ω a bounded open subset of Ω. We define OL p (Ω, V ) to be the space of holomorphic functions on Ω such that V |f | p dxdy < ∞, see [2], [4].…”