Tangential Limits for Harmonic Functions with Respect to ϕ(Δ): Stable and Beyond

In this paper, we discuss tangential limits for regular harmonic functions with respect to ϕ(Δ):= −ϕ(−Δ) in the C1,1 open set D in ℝd, where ϕ is the complete Bernstein function and d ≥ 2. When the exterior function f is local Lp-Hölder continuous of order β on Dc with p ∈ (1, ∞] and β > 1/p, for a large class of Bernstein function ϕ, we show that the regular harmonic function uf with respect to ϕ(Δ), whose value is f on Dc, converges a.e. through a certain parabola that depends on ϕ and ϕ′. Our result includes the case ϕ(λ) = log(1 + λα/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.