A Generalized CR equation with isolated singularities

TL;DR

This paper investigates a generalized Cauchy-Riemann equation with isolated singularities, constructing solutions through integral operators and analyzing the existence of nontrivial solutions to the homogeneous case.

Contribution

It extends the theory of generalized CR equations by handling coefficients with isolated singularities and establishing solution existence via integral operator methods.

Findings

Solutions constructed using associated integral operators

Existence of nontrivial solutions for the homogeneous equation proven

Analysis of equations with coefficients having isolated singularities

Abstract

The generalized CR equation $u_{\bar{z}}=au+b\bar{u}+f$ is studied when the coefficients $a$ and $b$ have a finite number of singular points inside the domain. Solutions are constructed via the study of an associated integral operator and the existence of nontrivial solutions of the associated homogeneous equation is established.