A Liouville Theorem for the Fractional Laplacian

TL;DR

This paper generalizes the classical Liouville Theorem to the fractional Laplacian, showing that bounded alpha-harmonic functions in all of R^n are necessarily constant, extending fundamental harmonic analysis results.

Contribution

It provides a Liouville Theorem for the fractional Laplacian, a significant extension of classical harmonic function theory to non-local operators.

Findings

Alpha-harmonic functions bounded above or below are constant in R^n.

The result extends classical harmonic analysis to fractional Laplacians.

Provides theoretical foundation for non-local PDEs.

Abstract

We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove Every $\alpha$-harmonic function bounded either above or below in all of $R^n$ must be constant.