On the Integrability of the Maximal Ergodic Function

Abstract

Let G = Rd or Zd and consider an ergodic measure-preserving action of G on a probability space (X, X, P), let f E L'(X, P) and Mf be its maximal ergodic function. Our purpose is to prove the converse of the following theorem of N. Wiener: If I log' If l is integrable then Mf is integrable. For the particular case G = Z this result was already obtained by D. Ornstein whose proof is based on induced transformations and seems to be specific to Z, our proof is based on a result of E. M. Stein on the Hardy-Littlewood maximal function on Rd and its analogue on Zd. Introduction. Our main result is the if part of the following theorem (see the notations below). THEOREM. Let G = Rd or Zd with d > 1. Consider an ergodic measure-preserving action of G on a probability space (X, W, P), and let f be a positive integrable function on (X, X, P). The maximal function Mf is integrable and only f log+f is integrable. The if' part of the theorem is classical and was proved by N. Wiener (cf. [8], [2]), the if part was proved for the particular case G = Z by D. Ornstein (cf. [6], [5]), with a proof based on induced transformations. Our proof is quite different and is based on a result of E. M. Stein (cf. [7]) on the Hardy-Littlewood maximal function. Similar results on certain classes of martingales were proved by D. L. Burkholder (cf. [1]) and R. F. Gundy (cf. [4]). Let G = Rd or Zd with d > 1, let (X, 9X, P) be a probability space. We assume that G acts measurably by measure-preserving transformations on (X, 9f, P): we denote this action by G x X 3 (g, x) -gx E X and write gx= gx. Let ji be the Lebesgue measure on G, V is a measurable subset of G, we write IVI = ,(V), and ,i(dg) = dg. If G = Rd, we denote by V' the ball of radius r centered at 0. If f is a measurable function on Rd, we define the Hardy-Littlewood maximal function Mf of f by Mf(g) r = sup v lf(g +h)Idh. Now f E L1(X, f, P), then the function G D g --f,(g) = f(gx) is ,u-integrable on compact subsets of G for almost every x. We define Received by the editors July 6, 1979. AMS (MOS) subject classifications (1970). Primary 28A65.