An L<SUP>p</SUP>-Inequality With Application to Ergodic Theory

In the last few years a large number of papers have appeared in the literature dealing with the weak convergence of iterates of contractions and strong convergence of their averages. The trend started with the Blum-Hanson Theorem [3]. The latest result in this direction is due to Akcoglu and Sucheston [ 1 ], [2]. The purpose of this paper is to give a simple, straightforward proof of the Adcoglu-Sucheston theorem; the present proof avoids approximation by finite-dimensional operators for which the contraction case is reduced to the invertible isometry case, and thus avoids altogether the application of Akcoglu's "Dilation Theorem". Let (X,F_,#) be a measure space and for 1 < p < o% let LP = LP(X,F,#) denote the usual Banach space. We write LP+= { fG LPIf•>0} . The following inequality was suggested to us by [ 1 ] (where a particular case of this inequality appears): The LP-Inequality. Let 1 < p < oo. Let f • L• g • LP+. Then for any 0 < e < 1 1 we have, with o• = (p 1) + p-l: (1) ffp-1 g d#•ellfll•+e IIgll• +e•-f f. gp-1 d#. PROOF: We may assume without loss of generality that g > O. Let 0 < •/< K