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• W2034965718 abstract "A well-known result states that for all bounded n-harmonic functions on the polydisc DEn the nontangential limits exist for (Lebesgue) 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 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.' Let Dn be the n-dimensional polydisc and w an n-harmonic function on IDn , i.e., a continuous real-valued function that is harmonic separately in each variable. The following is a well-known result concerning a class of those functions [8]. Suppose w is n-harmonic and bounded on Dn; then for Lebesgue every t ? Tn as z = (zi, ..., Zn) tends to t = (ti, ..., tn) where each Zjtj nontangentially (and to stress the condition independent of the other variables) w(z) converges to a real number. In terms of potential theoretical expectations it is natural to expect the result to extend to bounded quotients of positive n-harmonic functions. More precisely, suppose u > 0 is an nharmonic function on Dn that is represented as the integral of a finite Borel measure jui on T1n relative to the product of the Poisson kernels [4, 8]. Let w be any u-bounded (i.e., Iw/ul bounded) n-harmonic function-on D1n . It is natural to expect the nontangential limits of w/u to exist for iuu every element of Tn. The first main result of this paper will prove this expectation to be false. We give further a necessary and sufficient condition on u under which the result is valid for all u-bounded n-harmonic functions. With a slightly different perspective it is natural to ask if the nontangential limits exist everywhere on T n for quasi-bounded n-harmonic functions. We consider the problem in the general situation of u-quasi-bounded n-harmonic functions. We recall that a positive n-harmonic function is said to be u-quasi-bounded if it is the limit Received by the editors August 9, 1989 and, in revised form, December 27, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 31B25; Secondary 32A40, 3 1C99." @default.
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• W2034965718 date "1992-04-01" @default.
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• W2034965718 title "Polydiscs and nontangential limits" @default.
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• W2034965718 doi "https://doi.org/10.1090/s0002-9939-1992-1113640-7" @default.
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