On asymptotical expansions for certain singular integrals: 2-dimensional case

TL;DR

This paper investigates the asymptotic behavior of certain operators in pseudo differential equations on manifolds with borders, providing explicit representations using distribution theory and Fourier transforms.

Contribution

It introduces explicit asymptotic expansions for singular integrals in the 2D case within the context of pseudo differential operators on bordered manifolds.

Findings

Derived explicit representations for operators using distribution theory

Constructed limit distributions with Fourier transform, Dirac delta, and Cauchy-type integrals

Enhanced understanding of asymptotic behavior in 2D pseudo differential operators

Abstract

One discusses a problem of asymptotical behavior 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 a help of the Fourier transform, Dirac mass-function and its derivatives, and well-known distribution related to the Cauchy type integral.