Weakly p-Dunford Pettis sets in L1(μ, X)

Sets in Banach spaces that are mapped into norm compact sets by operators T : X -> l(p) (called weakly p-Dunford Pettis sets), for 1 < p < infinity, are studied in arbitrary Banach spaces X and in the space L-1 (mu, X) of Bochner integrable functions. Sufficient conditions for a subset of L-1 (mu, X) to be a weakly p-Dunford Pettis set are given. It is shown that if X* is an element of C-p, and K is a bounded and uniformly L-p-integrable subset of L-p(mu, X), then K is a weakly p-DP set in L-1 (mu, X).