Let R denote the real line. Let {T t } te^ be a measure preserving ergodic flow on a non atomic finite measure space (X, 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.