A note on a conjecture concerning boundary uniqueness

Abstract

Abstract. We consider the following conjecture (from Huang, et al): Let ∆ + denote the upper half disc in C and let γ = (−1, 1) (viewed as an interval in the real axis in C). Assume that F is a holomorphic function on ∆ + with continuous extension up to γ such that F maps γ into {| Im z| ≤ C| Re z|}, for some positive C. If F vanishes to infinite order at 0 then F vanishes identically.We show that given the conditions of the conjecture, either F ≡ 0 or there is a sequence in ∆ + , converging to 0, along which Im F/ Re F (defined where Re F = 0) is unbounded.

One-Sided Extendability and p-Continuous Analytic Capacities

“…We mention that the one-sided extendability of a function f : γ * → C is meant as the existence of a function F : G ∪ J → C which is holomorphic on the Jordan domain G, continuous on G ∪ J and such that on the arc J of γ * we have F| J = f | J . Such notions of one-sided extendability have been considered in [4] and the references therein, but in the present article and [2] it is, as far as we know, the first time where the phenomenon is proven to be rare. At the end of Sect.…”

“…In Sect. 4, we prove that extendability and real analyticity are rare phenomena in various spaces of functions on locally injective curves γ . For the real analyticity result, we assume that γ is analytic and the result holds in any C k (γ ), k ∈ {0, 1, 2, 3, .…”

Non extendability from any side of the domain of definition as a generic property of smooth or simply continuous functions on an analytic curve

“…We note that this is the first time it is proved that the phenomenon of holomorphic extension from one side is topologically rare. For one-sided holomorphic extensions we refer to [4] and at the articles contained at its bibliography.…”

“…A careful observation of the above facts enables us to prove that generically all functions in C ∞ (T ) are nowhere extendable from neither side of the unit circle T . We notice that extendability from one only side of the domain of definition has already been considered in [4] and the references therein. There one investigates sufficient conditions so that, if all derivatives of a function at a point of the boundary vanish one can conclude that the one side extension is identically equal to zero.…”