Noncommutative fractional integrals
Volume 229 / 2015
Abstract
Let $\mathcal M$ be a hyperfinite finite von Nemann algebra and $(\mathcal M_k)_{k\geq 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of $\mathcal M$. We investigate abstract fractional integrals associated to the filtration $(\mathcal M_k)_{k\geq 1}$. For a finite noncommutative martingale $x=(x_k)_{1\leq k\leq n} \subseteq L_1(\mathcal M)$ adapted to $(\mathcal M_k)_{k\geq 1}$ and $0 \lt \alpha \lt 1$, the fractional integral of $x$ of order $\alpha$ is defined by setting $$ I^\alpha x = \sum_{k=1}^n \zeta_k^{\alpha } dx_k $$ for an appropriate sequence $(\zeta_k)_{k\geq 1}$ of scalars. For the case of a noncommutative dyadic martingale in $L_1(\mathcal R)$ where $\mathcal R$ is the type ${\rm II}_1$ hyperfinite factor equipped with its natural increasing filtration, $\zeta_k=2^{-k}$ for $k\geq 1$.
We prove that $I^\alpha$ is of weak type $(1, 1/(1-\alpha))$. More precisely, there is a constant ${\mathrm c}$ depending only on $\alpha$ such that if $x=(x_k)_{k\geq 1}$ is a finite noncommutative martingale in $L_1(\mathcal M)$ then $$ \|I^\alpha x\|_{L_{1/(1-\alpha),\infty}({\mathcal M})}\leq {\mathrm c}\|x\|_{L_1(\mathcal M)}. $$ We also show that $I^\alpha$ is bounded from $L_{p}(\mathcal M)$ into $L_{q}(\mathcal M)$ where $1 \lt p \lt q \lt \infty$ and $\alpha=1/p-1/q$, thus providing a noncommutative analogue of a classical result. Furthermore, we investigate the corresponding result for noncommutative martingale Hardy spaces. Namely, there is a constant ${\mathrm c}$ depending only on $\alpha$ such that if $x=(x_k)_{k\geq 1}$ is a finite noncommutative martingale in the martingale Hardy space $\mathcal{H}_1(\mathcal M)$ then $\|I^\alpha x\|_{\mathcal{H}_{1/(1-\alpha)}(\mathcal M)}\leq {\mathrm c} \|x\|_{\mathcal{H}_1(\mathcal M)}$.
