ON THE ERGODIC AVERAGES AND THE ERGODIC HILBERT TRANSFORM

Let T be an invertible measure-preserving transformation on a sigma-fi nite measure space (X, mu) and let 1 < p < infinity. This paper uses a n abstract method developed by Jose Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the p ositive measurable functions nu such that the limit of the ergodic ave rages or the ergodic Hilbert transform exist for all f is an element o f L(p)(nu d mu). As a corollary, we obtain that both problems are equi valent, extending to this setting some results of R. Jajte, I. Berkson , J. Bourgain and A. Gillespie. We do not assume the boundedness of th e operator Tf(x) = f(Tx) on L(p)(nu d mu). However, the method of Rubi o de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the erg odic maximal operator and the ergodic Hilbert transform are bounded fr om L(p)(nu d mu) into L(p)(ud mu). We also study and solve the dual pr oblem.