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Distributional Boundary Values of Analytic Functions

Laboratory of Operator Theory and PDE: Foundations and Applications, Faculty of Exact Sciences, University of EL Oued , 39000, EL Oued, Algeria.

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Annals of Communications in Mathematics 2024
, 7 (1),
Received: 27 Dec 2023 |
Accepted: 15 Mar 2024 |
Published: 31 Mar 2024


Let D be a connected bounded domain in R2, S be its boundary which is closed, connected and smooth. Let Φ(z) = 1/2πi ∫S φ(s)ds/(s-z), φ ∈ X, z = x + iy, X is a Banach space of linear bounded functionals on Hμ, a Banach space of distributions, and Hμ is the Banach space of Hoelder-continuous functions on S with the usual norm. As X one can use also the space Hoelder continuous of bounded linear functionals on the Sobolev space H on S. Distributional boundary values of Φ(z) on S are studied in detail. The function Φ(t), t ∈ S, is defined in a new way. Necessary and sufficient conditions are given for φ ∈ X to be a boundary value of an analytic in D function. The Cauchy formula is generalized to the case when the boundary values of an analytic function in D are tempered distributions. The Sokhotsky-Plemelj formulas are derived for φ ∈ X.


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Distributional Boundary Values of Analytic Functions.

Annals of Communications in Mathematics,

7 (1):
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  • Copyright (c) 2023 by the Author(s). Licensee Techno Sky Publications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (


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