Sharp Green function estimates for $\Delta+ \Delta^{\alpha /2}$ in $C^{1,1}$ open sets and their applications

Abstract

We consider a family of pseudo differential operators $\{\Delta+ a^\alpha\Delta^{\alpha/2}$; $a\in[0, 1]\}$ on ${\mathbb R}^d$ that evolves continuously from $\Delta$ to $\Delta+ \Delta^{\alpha/2}$, where $d\geq1$ and $\alpha\in(0, 2)$. It gives rise to a family of Lévy processes $\{X^a, a\in[0, 1]\}$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green function of the process $X^a$ killed upon exiting a bounded $C^{1,1}$ open set $D\subset{\mathbb R}^d$. Our estimates are uniform in $a\in(0, 1]$ and taking $a\to0$ recovers the Green function estimates for Brownian motion in $D$. As a consequence of the Green function estimates for $X^a$ in $D$, we identify both the Martin boundary and the minimal Martin boundary of $D$ with respect to $X^a$ with its Euclidean boundary. Finally, sharp Green function estimates are derived for certain Lévy processes which can be obtained as perturbations of $X^a$.