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Carleson measures and vector-valued BMO martingales
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  • Published: 06 August 2008

Carleson measures and vector-valued BMO martingales

  • Yong Jiao1,2 

Probability Theory and Related Fields volume 145, pages 421–434 (2009)Cite this article

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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:

$$\widehat{\tau}=\{(w,k)\in \Omega\times \mathbb {N}: \tau(w)\leq k, \tau(w) < \infty\}.$$

We prove that there exists a positive constant c such that

$$\sup_{\tau}\frac{1}{P(\tau < \infty)}\int \limits_{{\widehat{\tau}}}\|df_k\|^qdP\otimes dm\leq c^q\|f\|_{BMO(X)}^q$$

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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Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques, Université de France-Comté, 25030, Besançon Cedex, France

    Yong Jiao

  2. School of Mathematics and Statistics, Wuhan University, 430072, Wuhan, China

    Yong Jiao

Authors
  1. Yong Jiao
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Corresponding author

Correspondence to Yong Jiao.

Additional information

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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Cite this article

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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  • Received: 31 March 2008

  • Revised: 14 July 2008

  • Published: 06 August 2008

  • Issue Date: November 2009

  • DOI: https://doi.org/10.1007/s00440-008-0173-7

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Keywords

  • Carleson measures
  • BMO martingales
  • Uniformly convex (smooth) spaces

Mathematics Subject Classification (2000)

  • Primary: 60G42
  • Secondary: 46L07
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