BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces

Ouyang, Caiheng ; Xu, Quanhua

Canadian journal of mathematics, Tome 62 (2010) no. 4, pp. 827-844

Résumé

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.

Détail

DOI : 10.4153/CJM-2010-043-6

Mots-clés : 46E40, 42B25, 46B20, BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces