Kernels of Integral Equations Can Be Boundedly Infinitely Differentiable on $\mathbb{R}^2$

TL;DR

This paper demonstrates that kernels of certain integral equations on  can be transformed into equivalent equations with infinitely differentiable kernels, facilitating analysis and solution methods.

Contribution

It introduces a method to reduce general third-kind integral equations to first or second kind with smooth kernels via unitary transformations.

Findings

Integral equations can be transformed into equivalent equations with smooth kernels.

The kernels are bi-Carleman and expandable in bilinear series.

The reduction simplifies the analysis of integral equations.

Abstract

In this paper, we reduce the general linear integral equation of the third kind in $L^2(Y,\mu)$, with largely arbitrary kernel and coefficient, to an equivalent integral equation either of the second kind or of the first kind in $L^2(\mathbb{R})$, with the kernel being the linear pencil of bounded infinitely differentiable bi-Carleman kernels expandable in absolutely and uniformly convergent bilinear series. The reduction is done by using unitary equivalence transformations.