The non-commutative Khintchine inequalities for p<1

We give a proof of the Khintchine inequalities in non-commutative $L_p$-spaces for all $0< p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for the analogues of such random variables in free probability. We also prove a factorization for operators from a Hilbert space to a non commutative $L_p$-space, which is new for $0<p<1$. We end by showing that Mazur maps are H\"older on semifinite von Neumann algebras. The main tool is a new form of H\"older inequality for non commutative Lp spaces with weights.