On Holomorphic Functions in the Upper Half-Plane Representable by Carleman Formula
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Complex Analysis and Operator Theory
On Holomorphic Functions in the Upper Half-Plane Representable by Carleman Formula Alekos Vidras1 Received: 23 March 2020 / Accepted: 14 August 2020 © Springer Nature Switzerland AG 2020
Abstract Let = {z = x + i y ∈ C : y > 0} be the upper half-plane and the interval [a, b] be a subset of ∂ = R. We derive a Carleman integral representation formula for all holomorphic functions f ∈ H() that have angular boundary values on [a, b] and which belong to the class N H1[a,b] (). The class N H1[a,b] () is the class of holomorphic functions in which belong to the Hardy class H1 near the interval [a, b] (“The Class of Functions Representable by Carleman Integral Representation Formula” section). As an application of the above characterization, our main result is an extension theorem for a function f ∈ L 1 ([a, b]) to a function f ∈ N H1[c,d] (), for almost all intervals [c, d] ⊂ (a, b). Similar results can be proved for a function f holomorphic in a slanted disc, with integrable boundary values on the horizontal part of the boundary. Keywords Carleman formules · Holomorphic extension from the part of the boundary Mathematics Subject Classification 30E20 · 30D25
1 Introduction Cauchy integral representation formula for a function f holomorphic in a simply connected domain D gives a way to evaluate the value of the function at z ∈ D from its values on the boundary ∂D. In a number of applications though, collecting the data
Communicated by Daniel Aron Alpay. This article is part of the topical collection “In memory of Carlos A. Berenstein (1944–2019)” edited by Irene Sabadini and Daniele Struppa.
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Alekos Vidras [email protected] Department of Mathematics and Statistics, University of Cyprus, POB 20537, 1678 Nicosia, Cyprus 0123456789().: V,-vol
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A. Vidras
(knowing the values of the function) over all of the boundary of the domain is not feasible, while only partial data (knowledge of boundary values on a part of the boundary) is available. Thus the need for integral representation formulas of holomorphic in a domain function from its knowledge on a part of the boundary of positive measure arises. Moreover, the converse problem becomes actual as well. Namely, knowing the function f on a part M ⊂ ∂D of positive measure, how one is able to decide that it has analytic continuation into the whole of D and that this analytic continuation belongs to particular class of analytic functions? In the present paper we study one particular case of the problems described above. Other known cases are described in [3–6]. Let = {z ∈ C : z > 0} be the upper half-plane. For the interval [a, b] ⊂ ∂, we consider the function i ψ(z) = π
b a
1 z−b dt = ln , z∈ z−t πi z − a
(1.1)
The real part ψ(z) of this function is harmonic and its value is equal to the angle that the point z ∈ “sees” the interval [a, b] divided by π . Thus, the geometric azb locus of the points ψ(z) = τ , τ > 0, is equal to the arc of the unique circumference t−b that passes through the poin
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