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 $1<q<\mathrm{\infty }$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$\underset{z_0\in D}{\sup }{\int }_D(1-\mathrm{\mid }z\mathrm{\mid }{)}^{q-1}\mathrm{\parallel }\mathrm{\nabla }f(z){\mathrm{\parallel }}^q{P}_{z_0}(z)dA(z)\le c^q\underset{z_0\in D}{\sup }{\int }_{\text{T}}\mathrm{\parallel }f(z)-f(z_0){\mathrm{\parallel }}^q{P}_{z_0}(z)dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-\mathrm{\mid }{z}_0{\mathrm{\mid }}^2}{\mathrm{\mid }1-\overline{z_0}z{\mathrm{\mid }}^2}\mathrm{.}$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.