Noncommutative fractional integrals

• Autorzy: Narcisse Randrianantoanina , Lian Wu
• Języki publikacji: EN

Abstrakty

EN

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(𝓡) where 𝓡 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 𝓗₁(ℳ) then $||I^{α}x||_{𝓗_{1/(1-α)}(ℳ)} ≤ c||x||_{𝓗₁(ℳ)}$.

Kategorie tematyczne

• 47A30: Norms (inequalities, more than one norm, etc.)
• 46L52: Noncommutative function spaces
• 46L53: Noncommutative probability and statistics
• 60G42: Martingales with discrete parameter
• 60G48: Generalizations of martingales

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2015
• Tom: 229
• Numer: 2
• Strony: 113-139

Daty

wydano

2015

Twórcy

autor

Narcisse Randrianantoanina

• Department of Mathematics, Miami University, Oxford, OH 45056, U.S.A.

autor

Lian Wu

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

Identyfikatory

DOI

10.4064/sm7989-1-2016