One-Sided Extendability and p-Continuous Analytic Capacities

Using complex methods combined with Baire’s Theorem, we show that one-sided extendability, extendability, and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-continuous analytic capacity and variants of it, p∈ { 0 , 1 , 2 , … } ∪ { ∞} , for compact or closed sets in C. We use these capacities in order to characterize the removability of singularities of functions in the spaces A ^p .