Noncommutative Fractional integrals

Abstract

Let $\M$ be a hyperfinite finite von Nemann algebra and $(\M_k)_{k\geq 1}$ be an increasing filtration of finite dimensional von Neumann subalgebras of $\M$. We investigate abstract fractional integrals associated to the filtration $(\M_k)_{k\geq 1}$. For a finite noncommutative martingale $x=(x_k)_{1\leq k\leq n} \subseteq L_1(\M)$ adapted to $(\M_k)_{k\geq 1}$ and $0<\alpha <1$, the fractional integral of $x$ of order $\alpha$ is defined by setting: $$I^{\alpha }x=\sum _{k=1}^n{\zeta }_k^{\alpha }d{x}_k$$ for an appropriate sequence of scalars $({\zeta }_k{)}_{k\ge 1}$. For the case of noncommutative dyadic martingale in $L_1(\mathbb{R})$ where $\mathbb{R}$ is the type ${\mathrm{II}}_1$ hyperfinite factor equipped with its natural increasing filtration, ${\zeta }_k=2^{-k}$ for $k\ge 1$. 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 $x=(x_k{)}_{k\ge 1}$ is a finite noncommutative martingale in $L_1(\M)$ then \[\|I^αx\|_{L_{1/(1-α),\infty}(\mathcal{\M})}\leq {\mathrm c}\|x\|_{L_1(\M)}.\] We also obtain that $I^{\alpha }$ is bounded from $L_{p}(\M)$ into $L_{q}(\M)$ where $1<p<q<\infty$ 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 $x=(x_k{)}_{k\ge 1}$ is a finite noncommutative martingale in the martingale Hardy space $\mathcal{H}_1(\M)$ then $\|I^αx\|_{\mathcal{H}_{1/(1-α)}(\M)}\leq {\mathrm c} \|x\|_{\mathcal{H}_1(\M)}$.