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 T, respectively. For 1 < q < ∞ and a Banach space B, we prove that there exists a positive constant c such that sup ∫(1- |Z|)^q-1||δ∫(Z)||qP _z0(Z) dA(z)≤c^q sup/_z0ΕD∫ _T||∫(z) - ∫(_z0)||qP_z0dm(z) holds for all trigonometric polynomials ∫ with coefficients in B if and only if B admits an equivalent norm which is q-uniformly convex, where P _z0(Z)=1-|^z0|2/|1-z0z|² The validity of the converse inequality is equivalent to the existence of an equivalent q-uniformly smooth norm.