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New definition of a singular integral operator

Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA.

* Corresponding Author
Annals of Communications in Mathematics 2023
, 6 (4),
Received: 13 September 2023 |
Accepted: 15 November 2023 |
Published: 31 December 2023


Let D be a connected bounded domain in R^2, S be its boundary which is closed, connected, and smooth, or S=(-∞,∞). Let Φ(z) be the function defined as Φ(z)=1/(2πi) ∫S(f(s)ds)/(s-z), where f∈L^1(S) and z=x+iy. The singular integral operator Af is defined as Af: =1/(iπ) ∫S(f(s)ds)/(s-t), where t∈S. This new definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f∈L^1(S) to be the boundary value of an analytic function in D. The Sokhotsky-Plemelj formulas are derived for f∈L^1(S). Our new definition allows one to treat singular boundary values of analytic functions.


Cite This Article

Alexander G. Ramm.
New definition of a singular integral operator.

Annals of Communications in Mathematics,

6 (4):

[1] Calderon A. Cauchy integral on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA, 74, N4, (1977), 1324-1327.
[2] Gahov F. Boundary value problems, Nauka, Moscow, (1977). (in Russian)
[3] Gel’fand I., Shilov G. Generalized functions, Vol. 1, Gos. Izdat. Fiz.-Math. Lit., Moscow, (1959). (In Russian)
[4] Mikhlin S., Pro¨ssdorf S. Singular integral operators, Springer-Verlag, New York, 1986.
[5] Muskhelishvili N. Singular integral equations, Nauka, Moscow, (1968). (In Russian)
[6] Ramm A.G. Boundary values of analytic functions, Far East Journal of Appl. Math., 116 (N3) (2023), 215-227.

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