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Noncommutative fractional integrals

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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 $||I^{α}x||_{𝓗_{1/(1-α)}(ℳ)} ≤ c||x||_{𝓗₁(ℳ)}$.
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Twórcy
  • Department of Mathematics, Miami University, Oxford, OH 45056, U.S.A.
autor
  • School of Mathematics and Statistics, Central South University, Changsha 410085, China
  • Department of Mathematics, Miami University, Oxford, OH 45056, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm7989-1-2016
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