On the Convergence of Poisson Integrals
Elias M. Stein, N. J. Weiss · Transactions of the American Mathematical Society · 1969
Introduction.The object of this paper is to extend two partial results, one positive and one negative, concerning the a.e.convergence of Poisson integrals on generalized half-planes equivalent to bounded symmetric domains.These results involve the distinction between "restricted" and "unrestricted" convergence, which already arose in the case of domains which are equivalent to product of half-planes (for this case see e.g.[Z, Chapter 17]).For these special domains there is restricted convergence for L", p^l, and unrestricted convergence for IS, p> 1.The L1 result of restricted convergence for various other tube domains was obtained more recently in [WJ and [S].The techniques set forth in [Wx] and [S] provide the starting point of our treatment here.Turning to the case of the general bounded symmetric domains, the positive results of [WJ and [W2] show that the Poisson integral of a function/on one of the domains in question converges restrictedly to/at a.e. point of the boundary if fie L",p> 1.It is demonstrated below that the condition/G L1 is also sufficient for restricted a.e.convergence, and that the Poisson integral of a measure has its Radon-Nikodym derivative as a restricted a.e.limit.On the other hand, it was essentially shown in [SWW] that for domains of the above type, there exists p0> 1 such that every LP class, p<p0, contains a function whose Poisson integral has oo as an unrestricted supremum at a.e. point of the boundary.The more complete result proved here is that pQ can be taken to be oo in every case, except rank one; and that there exists/gL00 with a Poisson integral which at a.e. point of the boundary fails to converge unrestrictedly to/.§2 is devoted to a few propositions of a more general nature which are necessary for the proof of the positive result in §3.The proof of the negative result is contained in §4.2. Real variable preliminaries.The results given here are necessary for estimates of the behavior of maximal averages over classes of sets which arise in §3.Note that the left Haar measure of a (measurable) subset £ of a locally compact group is denoted by |£|.The first auxiliary result, a covering theorem, generalizes results of Wiener [W] and Zygmund [Z, Chapter 17], and is proved by their method.See also [S].