Fine limits of superharmonic functions

Abstract

LetOdenote the origin of ℝn(wheren≥ 2), letB(X, r) denote the open ball in ℝnof centreXand radiusr, and define φn: (0, + ∞) → ℝ by φn(t) =t2-nifn≥ 3 andφ2(t) = log (1/t). A sequence (Xk) inB(O, 1) is said toconverge regularlytoOifXk→Oand there is a constantk> 1 such that φn(|Xk+1|) <Kφn(|Xk|) for eachk∈ ℕ. Ifn≥ 3, this is clearly equivalent to saying that there is a constantk′ ε (0, 1) such that |Xk+1| >k′|Xk| for eachk(cf. [18], p. 149).