Noncommutative Fractional integrals

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: $$I^\alpha x = \sum_{k=1}^n \zeta_k^{\alpha} dx_k$$ for an appropriate sequence of scalars $(\zeta_k)_{k\geq 1}$. For the case of noncommutative dyadic martingale in $L_1(\R)$ where $\R$ 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 $L_1(\M)$ then $$\|I^\alpha x\|_{L_{1/(1-\alpha),\infty}(\mathcal{\M})}\leq {\mathrm c}\|x\|_{L_1(\M)}.$$ 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 $x=(x_k)_{k\geq 1}$ is a finite noncommutative martingale in the martingale Hardy space $\mathcal{H}_1(\M)$ then $\|I^\alpha x\|_{\mathcal{H}_{1/(1-\alpha)}(\M)}\leq {\mathrm c} \|x\|_{\mathcal{H}_1(\M)}$.