The Fractional Malmheden Theorem

TL;DR

This paper introduces a fractional version of the Malmheden theorem, providing a new representation formula for s-harmonic functions and deriving optimal constants for the fractional Harnack inequality.

Contribution

It extends classical harmonic function results to the fractional setting, offering a novel representation formula and a new proof with optimal constants.

Findings

Representation formula for s-harmonic functions as superpositions of harmonic functions

New proof of the fractional Harnack inequality

Optimal constants for the fractional Harnack inequality in the ball

Abstract

We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.