Multilinear Marcinkiewicz-Zygmund Inequalities

We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on r -valued extensions of linear operators. We show that for certain 1 = p, q1,..., qm, r = 8, there is a constant C = 0 such that for every bounded multilinear operator T : Lq1(mu 1) x center dot center dot center dot xLqm (mu m). L p(.) and functions {f 1 k1} n1 k1= 1. Lq1(mu 1),..., {f m km} nm km= 1. Lqm (mu m), the following inequality holds parallel to k1km | T ( f 1 k1,, f m km)| r.. 1/ r L p(.) = C T m i= 1 ni ki= 1 | f i ki | r.. 1/ r Lqi (mu i). (1) In some cases we also calculate the best constant C = 0 satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderon-Zygmund operators.