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BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces


Published:20100520
Printed: Aug 2010
Caiheng Ouyang,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China
Quanhua Xu,
Laboratoire de Mathématiques, Université de FrancheComté, Besançon, France
Abstract
This paper studies the relationship between vectorvalued 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 $$\sup_{z_0\in D}\int_{D}(1z)^{q1}\\nabla f(z)\^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\f(z)f(z_0)\^qP_{z_0}(z) dm(z)$$ 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 $$P_{z_0}(z)=\frac{1z_0^2}{1\bar{z_0}z^2} .$$ 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
BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces
