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The boundary Harnack principle for the fractional Laplacian. (English) Zbl 0870.31009
The classical boundary Harnack principle on Lipschitz domains for the Laplacian \(\Delta\) on \(\mathbb{R}^n\), \(n\geq 2\), is extended to fractional powers \(\Delta^{\alpha/2}\), \(0<\alpha<2\) of the Laplacian, given by the generator of the symmetric stable semigroup of index \(\alpha\). Its corresponding potentials are known to be the Riesz potentials of order \(\alpha\). The main result is stated in Theorem 1:
Let \(\alpha \in (0,2)\) and \(n\geq 2\). Let \(D\) be a Lipschitz domain \(\mathbb{R}^n\) and \(V\) be an open set. For every compact set \(K\subset V\), there exists a positive constant \(C=C(\alpha, D,V,K)\) such that for all nonnegative functions \(u\) and \(v\) in \(\mathbb{R}^n\) which are continuous in \(V\), \(\alpha\)-harmonic in \(D\cap V\), vanish on \(D^c\cap V\), and satisfy \(u(x_0)= v(x_0)>0\) for some \(x_0 \in D\cap K\), we have \(C^{-1} u(x) \leq v(x) \leq Cu(x)\) for all \(x\in D \cap K\). Moreover, there exists a constant \(\eta= \eta (\alpha,D,V,K) >0\) such that the function \(u(x)/v(x)\) is Hölder continuous of order \(\eta\) in \(K\cap D\). In particular, for every \(Q\in \partial D \cap V\), \(\lim u(x)/v(x)\) exists as \(D\ni x\to Q\).
The article follows the arguments developed for elliptic operators by L. Caffarelli et al. [Indiana Univ. Math. J. 30, 621-640 (1981; Zbl 0512.35038)] as far as the major technical obstacle, the nonlocality of the fractional Laplacian, allows. It mainly uses probabilistic techniques related to the above symmetric stable process together with the explicit formulae of the “Poisson kernel” for balls and the Riesz potentials.
Reviewer: V.Metz (La Jolla)

MSC:
31B25 Boundary behavior of harmonic functions in higher dimensions
60J50 Boundary theory for Markov processes
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
60J99 Markov processes
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