engleski

Boundary Harnack principle for $\Delta + \Delta^{;; ; ; ; ; \alpha/2};; ; ; ; ; $

For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo differential operators $\{; ; ; ; ; ; \Delta+ b \Delta^{; ; ; ; ; ; \alpha/2}; ; ; ; ; ; ; b\in [0, 1]\}; ; ; ; ; ; $ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{; ; ; ; ; ; \alpha/2}; ; ; ; ; ; $. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to $\Delta +b \Delta^{; ; ; ; ; ; \alpha/2}; ; ; ; ; ; $ (or equivalently, the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with constant multiple $b^{; ; ; ; ; ; 1/\alpha}; ; ; ; ; ; $) in $C^{; ; ; ; ; ; 1, 1}; ; ; ; ; ; $ open sets. Here a ``uniform" BHP means that the comparing constant in the BHP is independent of $b\in [0, 1]$. Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to $\Delta + b \Delta^{; ; ; ; ; ; \alpha/2}; ; ; ; ; ; $ in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.

boundary Harnack principle ; harmonic function ; sub- and superharmonic function ; fractional Laplacian ; Laplacian ; symmetric $\alpha$-stable process ; Brownian motion ; Ito's formula ; Levy system ; martingales ; exit distribution

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