Structure of Cesaro function spaces'

The structure of the Cesaro function spaces Ces(p) on both [0. 1] and [0. infinity) for 1 < p <= infinity is investigated. We find their dual spaces, which equivalent norms have different description on [0, 1] and [0, infinity). The spaces Ces(p) for 1 < p < infinity are not reflexive but strictly convex. They are not isomorphic to any L-q space with 1 <= q <= infinity. They have "near zero" complemented subspaces isomorphic to l(P) and "in the middle" contain an asymptotically isometric copy of l(1) and also a copy of L-1[0, 1]. They do not have Dunford-Pettis property but they do have the weak Banach-Saks property Cesaro function spaces on [0. 1] and [0. infinity) are isomorphic for 1 < p <= infinity. Moreover, we give characterizations in terms of p and q when Ces(p)[0, 1] contains an isomorphic copy of l(q).