Noncommutative fractional integrals

Narcisse Randrianantoanina, and Lian Wu. "Noncommutative fractional integrals." Studia Mathematica 229.2 (2015): 113-139. <http://eudml.org/doc/285868>.

@article{NarcisseRandrianantoanina2015,
abstract = {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 $||I^\{α\}x||_\{_\{1/(1-α)\}(ℳ)\} ≤ c||x||_\{₁(ℳ)\}$.},
author = {Narcisse Randrianantoanina, Lian Wu},
journal = {Studia Mathematica},
keywords = {noncommutative probability; martingale transforms; fractional integrals; noncommutative martingale Hardy spaces},
language = {eng},
number = {2},
pages = {113-139},
title = {Noncommutative fractional integrals},
url = {http://eudml.org/doc/285868},
volume = {229},
year = {2015},
}

TY - JOUR
AU - Narcisse Randrianantoanina
AU - Lian Wu
TI - Noncommutative fractional integrals
JO - Studia Mathematica
PY - 2015
VL - 229
IS - 2
SP - 113
EP - 139
AB - 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 $||I^{α}x||_{_{1/(1-α)}(ℳ)} ≤ c||x||_{₁(ℳ)}$.
LA - eng
KW - noncommutative probability; martingale transforms; fractional integrals; noncommutative martingale Hardy spaces
UR - http://eudml.org/doc/285868
ER -