# On the asymptotic expansion of certain plane singular integral operators

On the asymptotic expansion of certain plane singular integral operators We discuss the problem of the asymptotic expansion for some operators in a general theory of pseudo-differential equations on manifolds with borders. Using the distribution theory one obtains certain explicit representations for these operators. These limit distributions are constructed with the help of the Fourier transform, the Dirac mass-function and its derivatives, and the well-known distribution related to the Cauchy type integral. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Boundary Value Problems Springer Journals

# On the asymptotic expansion of certain plane singular integral operators

, Volume 2017 (1) – Aug 15, 2017

## On the asymptotic expansion of certain plane singular integral operators

Chair of Diﬀerential Equations, We discuss the problem of the asymptotic expansion for some operators in a general Belgorod National Research University, Studencheskaya 14/1, theory of pseudo-diﬀerential equations on manifolds with borders. Using the Belgorod, 308007, Russia distribution theory one obtains certain explicit representations for these operators. These limit distributions are constructed with the help of the Fourier transform, the Dirac mass-function and its derivatives, and the well-known distribution related to the Cauchy type integral. Keywords: pseudo-diﬀerential operator; distribution; singularity; asymptotic expansion 1 Introduction In the theory of pseudo-diﬀerential equations the main diﬃculty is studying model op- erators in canonical domains according to a local principle. It shows that for a Fredholm property of a general pseudo-diﬀerential operator on a compact manifold one needs the invertibility of its local representatives in each point of a manifold [, ]. The author wrote many times on the nature of these local representatives, these are distinct in dependence on a point of a manifold. Each ‘singularity’ of a compact manifold (a half-space is a model situation for the smooth part of a boundary, cone for the conical point, wedge, etc.) cor- responds to a certain distribution, and a convolution operator with this distribution de- scribes a local representative of an initial pseudo-diﬀerential operator in a corresponding point of the manifold. All details can be found in [–]. But singularities can be of distinct dimensions and it is possible that such singularities of a low dimension can be obtained from analogous singularities of full dimension. This means we need to ﬁnd distributions for limit cases when some of the parameters of the singularities tend to zero. This approach was partially realized in [, ], and [] is devoted to multi-dimensional constructions. Our idea is the...

/lp/springer_journal/on-the-asymptotic-expansion-of-certain-plane-singular-integral-mbbx2aF144
Publisher
Springer Journals
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general
eISSN
1687-2770
D.O.I.
10.1186/s13661-017-0847-0
Publisher site
See Article on Publisher Site

### Abstract

We discuss the problem of the asymptotic expansion for some operators in a general theory of pseudo-differential equations on manifolds with borders. Using the distribution theory one obtains certain explicit representations for these operators. These limit distributions are constructed with the help of the Fourier transform, the Dirac mass-function and its derivatives, and the well-known distribution related to the Cauchy type integral.

### Journal

Boundary Value ProblemsSpringer Journals

Published: Aug 15, 2017

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