Tangential limits for harmonic functions with respect to ϕ(Δ): stable and beyond

Original · EN

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)=(1+λα/²). Our proofs use both the probabilistic and analytic methods.