Polydiscs and nontangential limits

GowriSankaran, Kohur

doi:10.1090/S0002-9939-1992-1113640-7

Polydiscs and nontangential limits
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Proc. Amer. Math. Soc. 115 (1992), 977-984 Request permission

Abstract:

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

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