Abstract
We study the relationship between vector-valued BMO martingales and Carleson measures. Let \({(\Omega,\mathcal {F} ,P)}\) be a probability space and 2 ≤ q < ∞. Let X be a Banach space. Given a stopping time τ, let \({\widehat{\tau}}\) denote the tent over τ:
We prove that there exists a positive constant c such that
for any finite martingale with values in X iff X admits an equivalent norm which is q-uniformly convex. The validity of the converse inequality is equivalent to the existence of an equivalent p-uniformly smooth norm. And then we also give a characterization of UMD Banach lattices.
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Partially supported by the Agence Nationale de Recherche (06-BLAN-0015), the National Natural Science Foundation of China (10371093) and China Scholarship Council (2007U13085).
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Jiao, Y. Carleson measures and vector-valued BMO martingales. Probab. Theory Relat. Fields 145, 421–434 (2009). https://doi.org/10.1007/s00440-008-0173-7
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DOI: https://doi.org/10.1007/s00440-008-0173-7