Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

TL;DR

This paper establishes the Martin representation and Relative Fatou Theorem for non-negative singular harmonic functions related to a fractional Laplacian with a gradient perturbation on smooth bounded domains.

Contribution

It extends classical potential theory results to a fractional Laplacian with a gradient term, providing new representation and boundary behavior theorems.

Findings

Martin representation for non-negative singular L-harmonic functions

Relative Fatou Theorem for fractional Laplacian with gradient perturbation

Results on ${ m C}^{1,1}$ bounded open sets

Abstract

Let $L=\Delta^{\alpha/2}+ b\cdot\nabla$ with $\alpha\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.