A noncommutative martingale convexity inequality

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful semifinite normal weight $\varphi$ and $\mathcal{N}$ be a von Neumann subalgebra of $\mathcal{M}$ such that the restriction of $\varphi$ to $\mathcal{N}$ is semifinite and such that $\mathcal{N}$ is invariant by the modular group of $\varphi$. Let $\mathcal{E}$ be the weight preserving conditional expectation from $\mathcal{M}$ onto $\mathcal{N}$. We prove the following inequality: $$\mathrm{\parallel }x{\mathrm{\parallel }}_p^2\ge \parallel \mathcal{E}(x){\parallel }_p^2+(p-1)\parallel x-\mathcal{E}(x){\parallel }_p^2,x\in L_p(\mathcal{M}),1<p\le 2,$$ which extends the celebrated Ball-Carlen-Lieb convexity inequality. As an application we show that there exists ${\epsilon }_0>0$ such that for any free group ${\mathbb{F}}_n$ and any $q\ge 4-{\epsilon }_0$, $$\mathrm{\parallel }{P}_t{\mathrm{\parallel }}_{2\to q}\le 1\iff t\ge \log \sqrt{q-1},$$ where $(P_t)$ is the Poisson semigroup defined by the natural length function of ${\mathbb{F}}_n$.