Boundedness of vector-valued martingale transforms on extreme points and applciations

Abstract

Let Bi, B2 be a pair of Banach spaces and T be a vector valued martingale transform (with respect to general filtration) which maps Bi-valued martingales into B 2 -valued martingales. Then, the following statements are equivalent: T is bounded from L p Ki into L^ for some p (or equivalently for every p) in the range 1 < p < 00; T is bounded from Ljjfj into BMO Kl ; T is bounded from BMO Bl into BMOn 2 ; T is bounded from H^ into H^. Applications to UMD and martingale cotype properties are given. We also prove that the Hardy space H^ defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if B has the Radon-Nikodym property.2000 Mathematics subject classification: primary 60G42; secondary 60B11,46B20.

Duality theorem for B‐valued martingale Orlicz–Hardy spaces associated with concave functions

“…In order to completely characterize the dual space of the Orlicz-Hardy space for B-valued martingales, it is more suitable to define the atoms as martingales. This idea is enlightened by Martínez and Torrea [13].…”

“…Atoms are usually defined as functions and so they are in the scalar-valued case. However, it does depend on the geometry of the underlying Banach space in the vector-valued case (see for example [9], [10], [13], [20]), as well as Theorem 3.8 states. In order to completely characterize the dual space of the Orlicz-Hardy space for B-valued martingales, it is more suitable to define the atoms as martingales.…”

“…In the study of the analytic properties of vector-valued martingale Hardy classes, it turned out that many definitions of the scalar-valued martingale Hardy spaces equivalent to the standard one are not equivalent to each other when the martingales are allowed to take values in a general Banach space B. Consequently, to obtain vector-valued analogues of some classical results it is necessary to impose restrictions on B or to develop new techniques. See for example the book of Liu [9] as well as the articles [7], [10], [12], [13], [15], [16], [19], [20], [21].…”

Embeddings between weak Orlicz martingale spaces

“…One main tool in our proofs will be martingale transforms with operator-valued multiplying sequences, defined and studied in [12,13] and [14]; now we give the definition. Definition 1.3.…”

Operator-valued martingale transform on mixed martingale Hardy spaces

“…In 2000, Martinez and Torrea [18] introduced the operator-valued martingale transforms and replaced the scalarvalued multiplying sequences by operator-valued multiplying sequences, which are a generalization of Burkholder's martingale transforms. Some further results and applications for operator-valued martingale transforms were developed in [19,17,13,12].…”