J. Aust. Math. Soc.  76 (2004), 207-221
Boundedness of vector-valued martingale transforms on extreme points and applications

Teresa Martínez
  Departamento de Matemáticas
  Universidad Autónoma de Madrid
  Ciudad Universitaria de Canto Blanco
  28049 Madrid
  Spain
  teresa.martinez@uam.es
and
José L. Torrea
  Departamento de Matemáticas
  Universidad Autónoma de Madrid
  Ciudad Universitaria de Canto Blanco
  28049 Madrid
  Spain
  joseluis.torrea@uam.es


Abstract
Let $\mathbf{B}_1$, $\mathbf{B}_2$ be a pair of Banach spaces and $T$ be a vector valued martingale transform (with respect to general filtration) which maps $\mathbf{B}_1$-valued martingales into $\mathbf{B}_2$-valued martingales. Then, the following statements are equivalent: $T$ is bounded from $L^p_{\mathbf{B}_1}$ into $L^p_{\mathbf{B}_2}$ for some $p$ (or equivalently for every $p$) in the range $1<p<\infty$; $T$ is bounded from $L^\infty_{\mathbf{B}_1}$ into $\textit{BMO}_{\mathbf{B}_2}$; $T$ is bounded from $\textit{BMO}_{\mathbf{B}_1}$ into $\textit{BMO}_{\mathbf{B}_2}$; $T$ is bounded from $H^1_{\mathbf{B}_1}$ into $H^1_{\mathbf{B}_2}$. Applications to $\textit{UMD}$ and martingale cotype properties are given. We also prove that the Hardy space $H^1_\mathbf{B}$ defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if $\mathbf{B}$ has the Radon-Nikodým property.
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