One-Sided Extendability and p-Continuous Analytic Capacities

Abstract

Using complex methods combined with Baire's Theorem, we show that onesided 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 .

Spherical arc-length as a global holomorphic parameter for analytic curves in the Riemann sphere

“…This led two of the authors of the present paper to prove that arc-length is a global conformal parameter for any analytic curve [7,8]. Thus, nothing changes in the results of [2,3] if we use the arc-length parametrization of the curve considered.…”

On Bergman type spaces of holomorphic functions and the density, in these spaces, of certain classes of singular functions

“…Thus the function (1) does not depend «essentially» on the choice of the defining function of  , as long as this set is convex with 1 C boundary. Notice also that the functions  f depend continuously on the point  .…”

“…The following theorem follows easily from Theorem (ii) of §2. See also [1], [4] and [8] . The last conclusion of the theorem follows from the well-known fact that the…”

Relations of the Spaces $${\varvec{A^p(\varOmega )}}$$ A p ( Ω ) and

“…. } ∪ {∞}, the derivatives f (l) , 0 ≤ l ≤ p, continuously extend over Ω ∪ J if and only if the continuous extension of f on J is p times continuously differentiable on J with respect to the position [2]. To do this, we place a smoothness assumption for the Riemann map φ : D → Ω from the open unit disk D onto Ω.…”