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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 $0<α<1$, the fractional integral of $x$ of order $α$ is defined by setting: $$I^αx = \sum_{k=1}^n ζ_k^α dx_k$$ for an appropriate sequence of scalars $(ζ_k)_{k\geq 1}$. 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, $ζ_k=2^{-k}$ for $k\geq 1$. We prove that $I^α$ is of weak-type $(1, 1/(1-α))$. More precisely, there is a constant ${\mathrm c}$ depending only on $α$ such that if $x=(x_k)_{k\geq 1}$ 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 $I^α$ is bounded from $L_{p}(\M)$ into $L_{q}(\M)$ where $1<p<q<\infty$ and $α=1/p-1/q$, 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 $α$ such that if $x=(x_k)_{k\geq 1}$ is a finite noncommutative martingale in the martingale Hardy space $\mathcal{H}_1(\M)$ then $\|I^αx\|_{\mathcal{H}_{1/(1-α)}(\M)}\leq {\mathrm c} \|x\|_{\mathcal{H}_1(\M)}$.