INTEGRAL REPRESENTATION OF QUANTUM MARTINGALES

A stochastic integral representation in terms of generalized integral kernel operator is proved for a wide class of quantum martingales which includes regular martingales and the martingales determined by the second quantization of integral operators containing singular kernels.

A Dimension-Free Weak-Type Estimate for Operators on UMD-Valued Functions

Let T denote the unit circle in the complex plane, and let X be a Banach space that satisfies Burkholder’s UMD condition. Fix a natural number, N ∈ . Let P denote the reverse lexicographical order on ZN. For each f ∈ L1(TN, X), there exists a strongly measurable function such that formally, for all . In this paper, we present a summation method for this conjugate function directly analogous to the martingale methods developed by Asmar and Montgomery-Smith for scalar-valued functions. Using a stochastic integral representation and an application of Garling’s characterization of UMD spaces, we prove that the associated maximal operator satisfies a weak-type (1, 1) inequality with a constant independent of the dimension N.