H\"older extension for fractional Laplacian

TL;DR

This paper characterizes the precise boundary conditions needed for fractional harmonic extensions to maintain H"older continuity up to the boundary, using estimates of fractional harmonic measure decay and domain fatness.

Contribution

It provides a sharp boundary condition characterization for fractional harmonic extensions with boundary H"older regularity, advancing understanding of fractional Laplacian boundary behavior.

Findings

Identifies the exact boundary conditions for H"older continuity of fractional harmonic extensions.

Establishes estimates of fractional harmonic measure decay.

Relates boundary regularity to fractional fatness of the domain's complement.

Abstract

In this note, we characterize the sharp boundary condition such that the fractional harmonic extensions with H\"older regularity up to the boundary is globally H\"older continuous. The proofs are based on estimates of fractional harmonic measure decay and uniform fractional fatness of the complement of the domain.