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

In this paper, we discuss tangential limits for regular harmonic functions with respect to I center dot(Delta):= -I center dot(-Delta) in the C (1,1) open set D in a"e (d) , where I center dot is the complete Bernstein function and d a parts per thousand yen 2. When the exterior function f is local L (p) -Holder continuous of order beta on D (c) with p a (1, a] and beta > 1/p, for a large class of Bernstein function I center dot, we show that the regular harmonic function u (f) with respect to I center dot(Delta), whose value is f on D (c) , converges a.e. through a certain parabola that depends on I center dot and I center dot ('). Our result includes the case I center dot(lambda) = log(1 + lambda (alpha/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.