A noncommutative martingale convexity inequality

Let M be a von Neumann algebra equipped with a faithful semifinite normal weight . and N be a von Neumann subalgebra of M such that the restriction of . to N is semifinite and such that N is invariant by the modular group of .. Let E be the weight preserving conditional expectation from M onto N. We prove the following inequality: .x.2p..E(x).2p+(p.1).x.E(x).2p,x.Lp(M),1<p.2, which extends the celebrated Ball.Carlen.Lieb convexity inequality. As an application we show that there exists .0>0 such that for any free group Fn and any q.4..0, .Pt.2.q.1.t.logq.1....., where (Pt) is the Poisson semigroup defined by the natural length function of Fn.