Polydiscs and nontangential limits

Abstract

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