On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circle

Abstract

AbstractLet D be a unit disk andM be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l (M) < 2π. Our first result characterizes the restriction of the holomorphic functions f ∈ ℋ︁(D), which are in the Hardy class ℋ︁1 near the arcM and are denoted by 𝒩 ℋ︁1M (𝒟), to the open arcM. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M.As an application of the above characterization, we present an extension theorem for a function f ∈ L 1(M) from any symmetric sub‐arc L ⊂ M of the unit circle, such that $ \bar L $ ⊂ M, to a function f ∈ 𝒩 ℋ︁1L (𝒟). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)