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

Abstract

For $d\ge 1$ 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 Lévy 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.

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