BMO functions and Carleson measures with values in uniformly convex spaces

Details BMO functions and Carleson measures with values in uniformly convex spaces BMO functions and Carleson measures with values in uniformly convex spaces

2007-05-14

arxiv.org/abs/0705.1948v2

Mathematics

Operator Algebras

To appear in Canadian J. Math

Scientific paper

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 $\mathbb T$, respectively. For $1< q<\infty$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\T}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ 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.

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