On a class of holomorphic functions representable by Carleman formulas in the interior of an equilateral cone from their values on its rigid base

Abstract

Let Δ be an equilateral cone in C with vertices at the complex numbers 0, z10, z20 and rigid base M (Section 1). Assume that the positive real semi-axis is the bisectrix of the angle at the origin. For the base M of the cone Δ we derive a Carleman formula representing all those holomorphic functions f ∈ H (Δ) from their boundary values (if they exist) on M which belong to the class NHM1 (Δ). The class NHM1 (Δ) is the class of holomorphic functions in Δ which belong to the Hardy class H1 near the base M (Section 2). As an application of the above characterization, an important result is an extension theorem for a function f ∈ L1 (M) to a function f ∈ NHM1 (Δ). © 2005 Elsevier Inc. All rights reserved.