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Article Dans Une Revue Canadian Journal of Mathematics = Journal Canadien de Mathématiques Année : 2010

BMO functions and Carleson measures with values in uniformly convex spaces

Résumé

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.

Dates et versions

hal-00477039 , version 1 (27-04-2010)

Identifiants

Citer

Caiheng Ouyang, Quanhua Xu. BMO functions and Carleson measures with values in uniformly convex spaces. Canadian Journal of Mathematics = Journal Canadien de Mathématiques, 2010, 62 (4), pp.827-844. ⟨10.4153/CJM-2010-043-6⟩. ⟨hal-00477039⟩
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