Multilinear Marcinkiewicz-Zygmund Inequalities

CARANDO, DANIEL; MAZZITELLI, MARTIN; OMBROSI, SHELDY

JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS

BIRKHAUSER BOSTON INC

Año: 2017

1069-5869

We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on ℓrℓr -valued extensions of linear operators. We show that for certain 1≤p,q1,?,qm,r≤∞1≤p,q1,?,qm,r≤∞ , there is a constant C≥0C≥0 such that for every bounded multilinear operator T:Lq1(μ1)×⋯×Lqm(μm)→Lp(ν)T:Lq1(μ1)×⋯×Lqm(μm)→Lp(ν) and functions {f1k1}n1k1=1⊂Lq1(μ1),?,{fmkm}nmkm=1⊂Lqm(μm){fk11}k1=1n1⊂Lq1(μ1),?,{fkmm}km=1nm⊂Lqm(μm) , the following inequality holds‖‖‖‖‖(∑k1,?,km|T(f1k1,?,fmkm)|r)1/r‖‖‖‖‖Lp(ν)≤C‖T‖∏i=1m‖‖‖‖‖(∑ki=1ni|fiki|r)1/r‖‖‖‖‖Lqi(μi).‖(∑k1,?,km|T(fk11,?,fkmm)|r)1/r‖Lp(ν)≤C‖T‖∏i=1m‖(∑ki=1ni|fkii|r)1/r‖Lqi(μi).(1)In some cases we also calculate the best constant C≥0C≥0 satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.