论文标题
DIRICHLET形式在边界上变性的尖锐的两侧绿色功能估计值
Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary
论文作者
论文摘要
在本文中，我们继续调查马尔可夫过程的潜在理论，而跳跃内核在边界处腐烂。更确切地说，我们考虑$ {\ mathbb r}^d _+$中的过程，带有$ {\ mathcal b}的跳跃内核（x，x，x，y）| x-y |^{ - d-d-α} $，并杀死电势$κ（x）= cx_d^）边界部分$ {\ MATHCAL B}（x，y）$可与三个术语的乘积与参数$β_1，β_2$，$β_3$和$β_4$相媲美。杀人术语中常数的$ c $可以作为$α$，$ {\ MATHCAL B} $的函数和（（（α-1）_+，α+β_1）$的函数，在$ p中严格增加至$ 0 $ p \ p \ p \ downarrof（$ p \ downarrow $ a $ p \ y lyfty as $ p \ y lyfty as $ p \ a+p \ y lyfty n $ p，$ p都在增加$ p \uparrowα+β_1$。我们对所有$ p \ in（（（α-1）_+，α+β_1）$的所有$ p \ y和$β_1，β_2$，$β_3$和$β_4$的所有可接受值建立了这些过程的绿色函数的尖锐的双向估计。根据$β_1$，$β_2$和$ P $所属的区域，绿色功能的估计值不同。实际上，根据参数所属的区域，估计值具有三种不同的形式。作为应用程序，我们证明边界harnack原理在参数的某些区域中存在，并且在参数的其他某些区域中失败。结合\ cite {ksv}的主要结果，我们完全确定了边界harnack原理所在的参数的区域。
In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${\mathbb R}^d_+$ with jump kernels of the form ${\mathcal B}(x,y) |x-y|^{-d-α}$ and killing potentials $κ(x)=cx_d^{-α}$, $0<α<2$. The boundary part ${\mathcal B}(x,y)$ is comparable to the product of three terms with parameters $β_1, β_2$, $β_3$ and $β_4$ appearing as exponents in these terms. The constant $c$ in the killing term can be written as a function of $α$, ${\mathcal B}$ and a parameter $p\in ((α-1)_+, α+β_1)$, which is strictly increasing in $p,$ decreasing to $0$ as $p\downarrow (α-1)_+$ and increasing to $\infty$ as $p\uparrowα+β_1$. We establish sharp two-sided estimates on the Green functions of these processes for all $p\in ((α-1)_+, α+β_1)$ and all admissible values of $β_1, β_2$, $β_3$ and $β_4$. Depending on the regions where $β_1$, $β_2$ and $p$ belong, the estimates on the Green functions are different. In fact, the estimates have three different forms depending on the regions the parameters belong to. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Combined with the main results of \cite{KSV},we completely determine the region of the parameters where the boundary Harnack principle holds.
