Noncommutative fractional integrals

Let M be a hyper finite 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) subset of L-1 (M) adapted to (M-k)(k >= 1) and 0 < alpha < 1, the fractional integral of x of order alpha is de fined by setting I-x(alpha) = Sigma(n)(k=1)zeta(alpha)(k)dx(k) for an appropriate sequence (zeta(k))(k >= 1) of scalars. For the case of a noncommutative dyadic martingale in L-1 (R) where R is the type II1 hyper finite factor equipped with its natural increasing filtration, zeta(k) = 2(-k) for k >= 1. We prove that I-alpha is of weak type (1; 1 = (1-alpha)). More precisely, there is a constant c depending only on alpha such that if x = (x(k)) k >= 1 is a finite noncommutative martingale in L-1 (M) then parallel to I(alpha)x parallel to(L1/(1-alpha),infinity)(M) <= c parallel to x parallel to(L1)(M). We also show that I-alpha is bounded from L-p (M) into L-q ( M) where 1 < p < q < infinity 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 c depending only on ff such that if x = (x(k))(k >= 1) is a finite noncommutative martingale in the martingale Hardy space H-1(M) then parallel to I(alpha)x parallel to(H1/(1-alpha))(M) <= c parallel to x parallel to(H1)(M).