Boundary Harnack principle for ∆+∆α/2
نویسندگان
چکیده
For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo differential operators {∆+b∆; b ∈ [0, 1]} on R that evolves continuously from ∆ to ∆ + ∆. 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∆ (or equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b) in C open sets. Here a “uniform” BHP means that the comparing constant in the BHP is independent of b ∈ [0, 1]. Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to ∆ + b∆ in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques. AMS 2000 Mathematics Subject Classification: Primary 31B25, 60J45; Secondary 47G20, 60J75, 31B05
منابع مشابه
A boundary Harnack principle in twisted Hölder domains
The boundary Harnack principle for the ratio of positive harmonic functions is shown to hold in twisted Hölder domains of order α for α ∈ (1/2, 1]. For each α ∈ (0, 1/2), there exists a twisted Hölder domain of order α for which the boundary Harnack principle fails. Extensions are given to L-harmonic functions for uniformly elliptic operators L in divergence form. Short title: Boundary Harnack ...
متن کاملBoundary Harnack Principle for Δ+δ
For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo differential operators {∆ + b∆α/2; b ∈ [0, 1]} on Rd that evolves continuously from ∆ to ∆ + ∆α/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∆α/2 (or equivalently, the sum of a Brownian motion and an independent ...
متن کاملEstimates and structure of α-harmonic functions
We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open sets D. This yields a unique representation of such functions as integrals against measures on Dc∪{∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test. 1 Main results and ...
متن کاملEquivalence between the Boundary Harnack Principle and the Carleson Estimate
Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this paper is to show that the boundary Harnack principle and the Carleson estimate are eq...
متن کاملBoundary Behavior of Harmonic Functions for Truncated Stable Processes
For any α ∈ (0, 2), a truncated symmetric α-stable process in R is a symmetric Lévy process in R with no diffusion part and with a Lévy density given by c|x| 1{|x|<1} for some constant c. In [24] we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this proc...
متن کاملA ug 2 00 9 Boundary Harnack principle for ∆ + ∆
For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo differential operators {∆+b∆α/2; b ∈ [0, 1]} on R that evolves continuously from ∆ to ∆ + ∆. 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∆ (or equivalently, the sum of a Brownian motion and an independent symmetric α...
متن کامل