2014/05/09 by Jaehoon Kang, Kang, Jaehoon, Panki Kim +1
Mathematics · #31B25 #60J75 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:31B25 #msc:60J75
paper · pdf · doi:10.48550/arxiv.1405.2141
17pages
arxiv created 2014/10/20 · arxiv updated 2014/10/21
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.