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BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces

Published online by Cambridge University Press:  20 November 2018

Caiheng Ouyang
Affiliation:
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China, e-mail: ouyang@wipm.ac.cn
Quanhua Xu
Affiliation:
Laboratoire de Mathématiques, Université de Franche-Comté, Besanҫon, France, e-mail: qxu@univ-fcomte.fr
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Abstract

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This paper studies the relationship between vector-valued $\text{BMO}$ 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 $\mathbb{T}$ , respectively. For $1\,<\,q\,<\,\infty $ and a Banach space $B$ , we prove that there exists a positive constant $c$ such that

$$\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{D}}{{\left( 1-\left| z \right| \right)}^{q-1}}{{\left\| \nabla f\left( z \right) \right\|}^{q}}{{P}_{{{Z}_{0}}}}\left( z \right)dA\left( z \right)\le {{c}^{q}}\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{\mathbb{T}}}{{\left\| f\left( z \right)-f\left( {{z}_{0}} \right) \right\|}^{q}}{{P}_{{{z}_{0}}}}\left( z \right)dm\left( z \right)$$

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

$${{P}_{{{z}_{0}}}}\left( z \right)=\frac{1-|{{z}_{0}}{{|}^{2}}}{|1-{{{\bar{z}}}_{0}}z{{|}^{2}}}.$$

The validity of the converse inequality is equivalent to the existence of an equivalent $q$ -uniformly smooth norm.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

References

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