Noncommutative Fractional integrals

TL;DR

This paper introduces and analyzes noncommutative fractional integrals associated with filtrations of von Neumann algebras, establishing their boundedness properties across various noncommutative Lp and Hardy spaces, extending classical fractional integral results.

Contribution

It defines noncommutative fractional integrals for martingales and proves their weak-type and boundedness properties, providing a noncommutative analogue of classical fractional integral theorems.

Findings

I^eta is of weak-type (1, 1/(1-eta)) for noncommutative martingales.

I^eta maps L_p(M) into L_q(M) with bounded norm, where 1/p - 1/q = eta.

Boundedness of I^eta on noncommutative martingale Hardy spaces.

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: $$I^\alpha x = \sum_{k=1}^n \zeta_k^{\alpha} dx_k$$ 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 $\|I^\alpha x\|_{\mathcal{H}_{1/(1-\alpha)}(\M)}\leq {\mathrm c} \|x\|_{\mathcal{H}_1(\M)}$.