Let f1, f2, . . . , fn be a family of independent random variables on a given probability space (Ω, µ). If 1 ≤ q ≤ p < ∞, the combination of Khintchine and Rosenthal inequalities easily produces the mixed-norm inequality

Xn k=1 fk ⊗ δk Lp(Ω;ℓq) ∼ Xn k=1 kfkk p p 1 p + Xn k=1 kfkk q q 1 q ,

with constants independent of n. We shall discuss noncommutative forms of this inequality and its dual version for 1 < p ≤ q ≤ ∞. Moreover, we shall introduce a transference method which allows us to replace freeness by a wide notion of noncommutative independence. These mixed-norm inequalities are in the core of recent results on operator space Lp embedding theory by M. Junge and the speaker. Our transference method provides better constants in some cases and a new nonembedability result. This is joint work with Marius Junge.