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 thatholds for all trigonometric polynomials f with coefficients in B if and only if B admits an equivalent norm which is q-uniformly convex, whereThe validity of the converse inequality is equivalent to the existence of an equivalent q-uniformly smooth norm.