New definition of singular integral operator

descriptionPublicationkeyboard_double_arrow_right Article 01 Jul 2023Publisher:Peertechz Publications Private LimitedJournal:Annals of Mathematics and Physics, volume 6, pages 97-99 (eissn: 2689-7636,

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Authors: Alexander G. Ramm;

doi: 10.17352/amp.000087 , 10.5281/zenodo.10445477 , 10.5281/zenodo.10445478

New definition of singular integral operator

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Abstract

Let D be a connected bounded domain in ℝ2, S be its boundary which is closed, connected and smooth or S = (-∞,∞). Let , f∈L1(S), z = x+iy. The singular integral operator , t∈S, is defined in a new way. This definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f∈L1(S) to be boundary value of an analytic in D function. The Sokhotsky-Plemelj formulas are derived for f∈L1(S). Our new definition allows one to treat singular boundary values of analytic functions.

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