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# BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces

Published:2010-05-20
Printed: Aug 2010
• Caiheng Ouyang,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China
• Quanhua Xu,
Laboratoire de Mathématiques, Université de Franche-Comté, Besançon, France
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## Abstract

This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.
 Keywords: BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces
 MSC Classifications: 46E40 - Spaces of vector- and operator-valued functions 42B25 - Maximal functions, Littlewood-Paley theory 46B20 - Geometry and structure of normed linear spaces

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