ON A PROBLEM OF DOOB ABOUT ANGULAR AND FINE CLUSTER VALUES
1985 年 37 巻 3 号 p. 307-321
詳細
1985 年 37 巻 3 号 p. 307-321
In 1965, Joseph L. Doob proved that if f is a superharmonic function on a half-space and if A(p) and F(p) denote respectively the angular and fine cluster sets of f at a boundary point p, then one of the following cases holds at almost every boundary point p.
(i) - ∞ \in F(p) \cap A(p) and F(p)\matrix{\subset\cr\
e\cr}A(p)
(ii) f does not have an angular limit at p but has a finite fine limit and an equal normal limit there.
(iii) f has a finite angular limit and an equal fine limit at p.
He then asked whether in (i) the set F(p) can be a proper subinterval of A(p) on a P set of positive measure.
In this note, we study this problem in the two dimensional case. We construct a Nevanlinna's function for which (i) holds for a countably dense set of boundary points. Our result is sharp in the sense that the P set cannot be improved to be of positive measure. It is not clear whether the construction is possible for any P set of measure zero.
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