  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.
