Noncommutative fractional integrals

Narcisse Randrianantoanina¹ ; Lian Wu²

¹ Department of Mathematics Miami University Oxford, OH 45056, U.S.A.
² School of Mathematics and Statistics Central South University Changsha 410085, China and Department of Mathematics Miami University Oxford, OH 45056, U.S.A.

Studia Mathematica, Tome 229 (2015) no. 2, pp. 113-139

Résumé

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 $\|I^\alpha x\|_{\mathcal{H}_{1/(1-\alpha)}(\mathcal M)}\leq {\mathrm c} \|x\|_{\mathcal{H}_1(\mathcal M)}$.

Détail

DOI : 10.4064/sm7989-1-2016

Keywords: mathcal hyperfinite finite von nemann algebra mathcal geq increasing filtration finite dimensional von neumann subalgebras mathcal investigate abstract fractional integrals associated filtration mathcal geq finite noncommutative martingale leq leq subseteq mathcal adapted mathcal geq alpha fractional integral order alpha defined setting alpha sum zeta alpha appropriate sequence zeta geq scalars noncommutative dyadic martingale mathcal where mathcal type hyperfinite factor equipped its natural increasing filtration zeta k geq prove alpha weak type alpha precisely there constant mathrm depending only alpha geq finite noncommutative martingale mathcal alpha alpha infty mathcal leq mathrm mathcal alpha bounded mathcal mathcal where infty alpha p providing noncommutative analogue classical result furthermore investigate corresponding result noncommutative martingale hardy spaces namely there constant mathrm depending only alpha geq finite noncommutative martingale martingale hardy space mathcal mathcal alpha mathcal alpha mathcal leq mathrm mathcal mathcal