Noncommutative fractional integrals

Narcisse Randrianantoanina ; Lian Wu

Studia Mathematica, Tome 231 (2015), p. 113-139 / Harvested from The Polish Digital Mathematics Library

Résumé

Let ℳ be a hyperfinite finite von Nemann algebra and $(ℳ_{k})_{k≥1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration $(ℳ_{k})_{k≥1}$. For a finite noncommutative martingale $x = (x_{k})_{1≤k≤ n} ⊆ L₁(ℳ)$ adapted to $(ℳ_{k})_{k≥1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{α}x = ∑_{k=1}^{n} ζ_{k}^{α} dx_{k}$ for an appropriate sequence $(ζ_{k})_{k≥1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor equipped with its natural increasing filtration, $ζ_{k} = 2^{-k}$ for k ≥ 1. We prove that $I^{α}$ is of weak type (1,1/(1-α)). More precisely, there is a constant c depending only on α such that if $x = (x_{k})_{k≥1}$ is a finite noncommutative martingale in L₁(ℳ) then $||I^{α}x||_{L_{1/(1-α),∞}(ℳ)} ≤ c||x||_{L₁(ℳ)}$. We also show that $I^{α}$ is bounded from $L_{p}(ℳ)$ into $L_{q}(ℳ)$ where 1 < p < q < ∞ 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 ${c}$ depending only on α such that if $x = (x_{k})_{k≥1}$ is a finite noncommutative martingale in the martingale Hardy space ₁(ℳ) then $||I^{α}x||_{_{1/(1-α)}(ℳ)} ≤ c||x||_{₁(ℳ)}$.

Détail

Publié le : 2015-01-01

Zbl 06526964

EUDML-ID : urn:eudml:doc:285868