On a conformally invariant integral equation involving Poisson kernel

TL;DR

This paper investigates a unique conformally invariant integral equation involving the Poisson kernel on the unit ball, addressing existence issues with symmetry considerations and establishing solutions under antipodal symmetry constraints.

Contribution

It introduces the study of a non-PDE conformally invariant integral equation, proving existence results within antipodal symmetry classes.

Findings

Existence of solutions under antipodal symmetry.

Identification of Kazdan-Warner type obstructions.

The integral equation is not dual to standard PDEs.

Abstract

We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a Kazdan-Warner type obstruction to existence of solutions. We prove existence in the antipodal symmetry functions class.