Characterization of uniformly convex and smooth Banach spaces by using Carleson measures in Bessel settings (Q2850732)

\textit{C.-H. Ouyang} and \textit{Q.-H. Xu} [Can. J. Math. 62, No. 4, 827--844 (2010; Zbl 1206.46011)] showed that the fact that a Banach space has an equivalent \(q\)-uniformly convex norm can be described in terms of Carleson measures associated to Poisson integrals of vector-valued functions in \(BMO\). In the present paper, the authors make a parallel study when the standard Poisson semigroup is replaced by the Poisson semigroup \(P^\lambda_t\) associated to the Bessel operator \(\Delta_\lambda\). Their main result establishes that, for \(q\geq 2\), a Banach space \(E\) has an equivalent norm which is \(q\)-uniformly convex iff, for any odd function in \(BMO(\mathbb R, E)\), the measure \(d\mu_f(x,t)= \|t \partial_t P^\lambda_t(f)(x)\|^q\frac{dx\,dt}{t}\) is a Carleson measure on \((0,\infty)\times(0,\infty)\). The corresponding result for the dual notion of the \(q\)-uniformly smooth norm is also shown.