Multi-indexed $p$-orthogonal sums in non-commutative Lebesgue spaces Javier Parcet 46L5205A18Khintchine inequalityMoebius inversion$p$-orthogonal sums In this paper we extend a recent Pisier's inequality for $p$-orthogonal sums in non-commutative Lebesgue spaces. To that purpose, we generalize the notion of $p$-orthogonality to the class of multi-indexed families of operators. This kind of families appears naturally in certain non-commutative Khintchine type inequalities associated with free groups. Other $p$-orthogonal families are given by the homogeneous operator-valued polynomials in the Rademacher variables or the multi-indexed martingale difference sequences. As in Pisier's result, our tools are mainly combinatorial. Indiana University Mathematics Journal 2004 text pdf 10.1512/iumj.2004.53.2576 10.1512/iumj.2004.53.2576 en Indiana Univ. Math. J. 53 (2004) 1171 - 1188 state-of-the-art mathematics http://iumj.org/access/