Polydiscs and nontangential limits

Abstract

Abstract.A well-known result states that for all bounded «-harmonic functions on the polydisc D" the nontangential limits exist for (Lebesgue) almost every element of the «-torus. In this paper it is shown that a similar result is not in general valid for bounded quotients of two positive «-harmonic functions. Necessary and sufficient conditions on a «-harmonic function u > 0 are given to ensure the existence "almost everywhere" of the nontangential limits of the quotients w/u in the case (i) for all «-harmonic functions w suchthat w/u is bounded and in the case (ii) for all «-harmonic functions w that are ' «-quasi-bounded.'