On Fine Limits and Curving-Shaped Limits

Abstract

This paper studies the boundary behavior of the so-called SIH-functions, i.e., the functions satisfying the scale invariant Harnack inequality on a domain D ⊂ R^N (N ≥ 2). Suppose that D contains a curving-cone at a point ζ ∈ ∂D and u is a SIH-function on D. Then u has a curving-shaped limit L at ζ, if u has a Φ-fine limit (especially, a p-fine limit in the sense of [6] or an α-fine limit in the sense of the Riesz potential theory [4]) L at ζ.