Boundary Harnack principle for $\Delta + \Delta^{\alpha/2}$

TL;DR

This paper proves a uniform boundary Harnack principle for a family of operators interpolating between the Laplacian and a fractional Laplacian, with explicit decay rates, applicable to harmonic functions in smooth domains.

Contribution

It establishes a uniform boundary Harnack principle with explicit decay rates for operators combining Laplacian and fractional Laplacian, valid across a range of parameters.

Findings

Uniform boundary Harnack principle with explicit decay rates.

Carleson type estimate for harmonic functions in Lipschitz domains.

Method combines probabilistic and analytic techniques.

Abstract

For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to $\Delta +b \Delta^{\alpha/2}$ (or equivalently, the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with constant multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets. Here a "uniform" BHP means that the comparing constant in the BHP is independent of $b\in [0, 1]$. Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to $\Delta + b \Delta^{\alpha/2}$ in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.