$Δ+ Δ^{α/2}$在$C^{1,1}$开集中的Sharp格林函数估计及其应用
Sharp Green Function Estimates for $Δ+ Δ^{α/2}$ in $C^{1,1}$ Open Sets and Their Applications
摘要：
本文考虑$\R^d$上的一族伪微分算子$\{Δ+ a^αΔ^{α/2}; a\in [0, 1]\}$，它从$Δ$连续演化到$Δ+ Δ^{α/2}$，其中$d\geq 1$和$α\in (0, 2)$ .它产生了一族Lévy过程\{ $X^a, a\in [0, 1]\}$，其中$X^a$是布朗运动和一个独立的具有权$a$的对称$α$ -稳定过程之和。利用最近得到的一致边界Harnack原理与明确的衰减率，我们建立了尖锐的边界的格林函数的过程$X^a$退出有界$C^{1,1}$开集$D\subset\R^d$。因此，我们确定了$D$的马丁边界相对于$X^a$的欧几里得边界。最后，对于某些Lévy过程，得到了精确的格林函数估计，这些估计可以作为PDMATHAMT的扰动而得到.
英文摘要：
We consider a family of pseudo differential operators $\{Δ+ a^αΔ^{α/2}; a\in [0, 1]\}$ on $\R^d$ that evolves continuously from $Δ$ to $Δ+ Δ^{α/2}$, where $d\geq 1$ and $α\in (0, 2)$. 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 $α$-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 $D\subset\R^d$. As a consequence, we identify the 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 $X^a$.
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