Naslov
Sharp Green Function Estimates for $\Delta + \Delta^{;;; ; ; \alpha/2};;; ; ; $ in $C^{;;; ; ; 1, 1};;; ; ; $ Open Sets and Their Applications
(Sharp Green function estimates for Δ+Δα/2 in C1,1 open sets and their applications)

Autori
Chen, Zhen-Qing ; Kim, Panki ; Song, Renming ; Vondraček, Zoran

Izvornik
Illinois journal of mathematics (0019-2082) 54 (2010), 3; 981-1024

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Green function estimates ; boundary Harnack principle ; harmonic functions ; fractional Laplacian ; symmetric $\alpha$-stable process ; Brownian motion ; perturbation

Sažetak
We consider a family of pseudo differential operators $\{; ; ; ; ; \Delta+ a^\alpha \Delta^{; ; ; ; ; \alpha/2}; ; ; ; ; ; \ a\in [0, 1]\}; ; ; ; ; $ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{; ; ; ; ; \alpha/2}; ; ; ; ; $, where $d\geq 1$ and $\alpha \in (0, 2)$. It gives rise to a family of L\'evy 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 $D\subset\R^d$. Our estimates are uniform in $a\in (0, 1]$ and taking $a\to 0$ 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 L\'evy processes which can be obtained as perturbations of $X^a$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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