The limit case of the Cesaro-alpha convergence of the ergodic averages andthe ergodic Hilbert transform

Sarrion and the authors gave a sufficient condition on invertible Lamperti operators on L-P which guarantees the convergence in the Cesaro-alpha sense of the ergodic averages and the ergodic Hilbert transform for all f is an element of L-p with p > 1/(1 + alpha) and -1 < alpha less than or equal to 0. The result does not hold for the space L1/(1 + alpha). In this paper we give a positive result for the smaller Lorentz; space L-1/(1 + alpha),L-1.