Dirichlet Heat Kernel Estimates for $\Delta ^{\alpha /2}+ \Delta ^{\beta /2}$

Abstract

For $d\geq 1$ and $0<\beta<\alpha<2$, consider a family of pseudo differential operators $\{\Delta^{\alpha} + a^\beta \Delta^{\beta/2}; a \in [0, 1]\}$ that evolves continuously from $\Delta^{\alpha/2}$ to $ \Delta^{\alpha/2}+ \Delta^{\beta/2}$. It gives arise to a family of L\'evy processes \{$X^a, a\in [0, 1]\}$, where each $X^a$ is the sum of independent a symmetric $\alpha$-stable process and a symmetric $\beta$-stable process with weight $a$. For any $C^{1,1}$ open set $D$, we establish explicit sharp two-sided estimates (uniform in $a\in [0,1]$) for the transition density function of the subprocess $X^{a, D}$ of $X^a$ killed upon leaving the open set $D$. The infinitesimal generator of $X^{a, D}$ is the non-local operator $\Delta^{\alpha} + a^\beta \Delta^{\beta/2}$ with zero exterior condition on $D^c$. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for $X^{a, D}$ and uniform boundary Harnack principle for $X^a$ in $D$ with explicit decay rate.