Iterated fine limits and iterated nontangential limits

Abstract:

Let ${\Omega _k},k = 1{\text { to }}n$, be harmonic spaces of Brelot and ${u_k} > 0$ harmonic functions on ${\Omega _k}$. For each $w$ in a class of multiply superharmonic functions it is shown that the iterated fine limits of $[w/{u_1} \cdots {u_n}]$ exist up to a set of measure zero for the product of the canonical measures corresponding to ${u_k}$ and are independent of the order of iteration. This class contains all positive multiply harmonic functions on the product of ${\Omega _k}$’s. For a holomorphic function $f$ in the Nevanlinna class of the polydisc ${U^n}$, it is shown that the $n$th iterated fine limits exist and equal almost everywhere on ${T^n}$ the $n$th iterated nontangential limits of $f$, for any fixed order of iteration. It is then deduced that, with the exception of a set of measure zero on ${T^n}$, the absolute values of the different iterated limits of $f$ are equal. It is also shown that the $n$th iterated nontangential limits are equal almost everywhere on ${T^n}$ for any $f$ in ${N_1}({U^n})$.