Sharp tangent inequalities in noncommutative $L_p$-spaces

RESUMO

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite weight, and let $L_p(\mathcal{M})$ denote the associated Haagerup noncommutative $L_p$-space. For $1<p<\infty$, let $q=p/(p-1)$. For $z\in L_q(\mathcal{M})$ and $y\in L_p(\mathcal{M})$, denote by $\langle z,y\rangle=\operatorname{Re}\operatorname{Tr}(z^*y)$ the canonical real duality pairing between $L_q(\mathcal{M})$ and $L_p(\mathcal{M})$. Define the normalized duality map by $J_p(x)=\|x\|_p^{\,2-p}u|x|^{p-1}$ for $x=u|x|\neq0$, and set $J_p(0)=0$. We establish the sharp tangent inequality $$\|x+y\|_p^2\ge\|x\|_p^2+2\langle J_p(x),y\rangle+(p-1)\|y\|_p^2,\qquad x,y\in L_p(\mathcal{M}),\quad 1<p\le2.$$ The reverse inequality holds for $2\le p<\infty$, and the constant $p-1$ is optimal. Our inequality extends Xu's tangent inequality \cite{Xu89} from classical $L_p$-spaces to general noncommutative $L_p$-spaces and provides the sharp tangent formulation of the Ricard--Xu convexity inequality \cite{RX16}. When specialized to Schatten $p$-classes, it also yields the corresponding tangent formulation of the Ball--Carlen--Lieb convexity inequality \cite{BCL94}. Under a suitable positivity assumption, we further establish a sharp complement to Xu's tangent inequality for Schatten $p$-classes in the range $0<p<1$.