Harmonic functions on balls for x-dependent rectilinear stable processes

TL;DR

This paper derives precise estimates for harmonic functions related to x-dependent rectilinear stable processes within balls, utilizing barrier functions for the fractional Laplacian to handle variable dependence.

Contribution

It introduces a method to obtain sharp estimates for harmonic functions of x-dependent processes using barrier functions, advancing understanding of variable coefficient fractional operators.

Findings

Established sharp estimates for harmonic functions in balls

Developed barrier functions for x-dependent fractional Laplacian

Provided a framework for analyzing variable coefficient stable processes

Abstract

We obtain sharp estimates for functions harmonic with respect to $x$-dependent rectilinear stable processes in balls, under the assumption that the Dirichlet exterior data are radial about the center. The main idea of the proof is based on the construction of global barrier functions for the $x$-dependent rectilinear fractional Laplacian in balls.