Abstract
In this paper, we discuss tangential limits for regular harmonic functions with respect to ϕ(Δ):= −ϕ(−Δ) in the C 1,1 open set D in ℝ d, where ϕ is the complete Bernstein function and d ≥ 2. When the exterior function f is local L p-Hölder continuous of order β on D c with p ∈ (1, ∞] and β > 1/p, for a large class of Bernstein function ϕ, we show that the regular harmonic function u f with respect to ϕ(Δ), whose value is f on D c, converges a.e. through a certain parabola that depends on ϕ and ϕ ′. Our result includes the case ϕ(λ) = log(1 + λ α/2). Our proofs use both the probabilistic and analytic methods. In particular, the Poisson kernel estimates recently obtained in Kang and Kim (J. Korean Math. Soc. 50(5), 1009–1031, 2013) are essential to our approach.
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This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No.2009-0083521). This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (2013004822)
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Kang, J., Kim, P. Tangential Limits for Harmonic Functions with Respect to ϕ(Δ): Stable and Beyond. Potential Anal 42, 629–644 (2015). https://doi.org/10.1007/s11118-014-9449-y
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DOI: https://doi.org/10.1007/s11118-014-9449-y
Keywords
- Bernstein function
- Subordinate Brownian motion
- Poisson kernel
- Harmonic function
- (non) tangential limits
- L p-Hölder space