Boundary Harnack principle for Δ+Δ^{/2}

By Zhen-Qing Chen, Panki Kim, Renming Song, Zoran Vondraček. 2012

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For d � 1 and � 2 (0,2), consider the family of pseudo differential operators f �+b� �/2 ;b 2 (0,1)g on R d that evolves continuously fromto � + � �/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 �+b� �/2 (or equivalently, the sum of a Brownian motion and an independent symmetric�-stable process with constant multipleb 1/� ) inC 1,1 open sets. Here a "uniform" BHP means that the comparing constant in the BHP is independent of b 2 (0,1). Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to � +b� �/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/0a2e2d47b530f903029d936f361168f93fca735b

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Citations in the corpus

Potential theory of subordinate Brownian motions revisited (2011, background): “ open subset of Rd. The main results include the boundary Harnack principle and sharp Green function estimates. For processes having a continuous component see [26] (for the one-dimensional case) and [12, 13, 14] (for multi-dimensional case). For purely discontinuous processes, the boundary Harnack ”
Regularity results for solutions of mixed local and nonlocal elliptic equations (2022, background): “Moreover, mixed operators have recently received a great attention from different points of view, including regularity theory [2, 3, 8, 10, 12, 14, 15, 17, 25], existence and non-existence results [1, 6, 28], eigenvalue problems [13, 16, 22], shape optimization and calculus of variations [7, 11], sy”
Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation (2010, result): “Note that, by the same proof as that of [11], Lemma 4.”
Two‐sided Green function estimates for killed subordinate Brownian motions (2010, background): “Let V denote the potential measure of the ladder height process of Z.”