A Fatou theorem for $F$-harmonic functions

TL;DR

This paper establishes a Fatou-type theorem for $F$-harmonic functions on negatively curved manifolds, demonstrating their boundary convergence properties and uniqueness, with implications for equidistribution problems.

Contribution

It introduces a Fatou theorem for $F$-harmonic functions, extending classical harmonic analysis to a new class of functions with integral representations.

Findings

Proves nontangential convergence of quotients of $F$-harmonic functions.

Establishes uniqueness of $F$-harmonic functions on compact manifolds.

Provides an integral representation analogous to the Poisson formula.

Abstract

In this paper we study a class of functions that appear naturally in some equidistribution problems and that we call $F$-harmonic. These are functions of the universal cover of a closed and negatively curved which possess an integral representation analogous to the Poisson representation of harmonic functions, where the role of the Poisson kernel is played by a H\"older continuous kernel. More precisely we prove a theorem \`a la Fatou about the nontangential convergence of quotients of such functions, from which we deduce some basic properties such as the uniqueness of the $F$-harmonic function on a compact manifold and of the integral representation of $F$-harmonic functions.