BMO functions and Carleson measures with values in uniformly convex spaces

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 $$sup_{z_0in D}int_{D}(1-|z|)^{q-1}| abla f(z)|^q P_{z_0}(z) dA(z) le c^qsup_{z_0in D}int_{T}|f(z)-f(z_0)|^qP_{z_0}(z) dm(z)$$ 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.