On certain maximal functions and approach regions

Abstract

If f E L’(iR”) and u(x, y) is the Poisson integral of J; a classical theorem of Fatou [2] asserts that u has nontangential limits almost everywhere on R”. Littlewood [5], and later Zygmund [9], showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially. In this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded. It then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of Fatou’s theorem remain true. Suppose Q c IR:+ ’ has the property that whenever (xi, u,) E B and whenever [x -x, [ < y y, it follows that (x, v) E Q. Then the maximal function associated to such a set is bounded if and only if the cross-sectional measure at height y is bounded by a constant times y”. We also study regions where the cross-sectional area is larger than y”. We introduce a modified maximal function which is again bounded. This result leads to a simple proof of the boundedness of certain tangential maximal functions of Poisson integrals of potentials studied in 161. The plan of this paper is as follows: In Section 1, we motivate our main ideas by discussing two particular examples. The method of proof here is somewhat different from the proofs of the more general results. In Section 2, we obtain the necessary and sufficient conditions for Fatou’s theorem. In Section 3, we study more general approach regions, and the appropriately modified maximal functions. In Section 4, we show how this maximal function controls the tangential boundary behavior of Poisson integrals of potentials.