We obtain new characterizations of the q-uniformly convex and smooth Banach spaces by using Carleson measures. These measures are defined by Poisson integral associated with Bessel operators and Banach valued BMO-functions. By the way we describe q-uniformly convexity and smoothness of a Banach space in terms of the mapping properties of the Lusin integral defined by the Poisson semigroup for Bessel operators.

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Jorge J. Betancor

Dep. de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez s/n, 38271 La Laguna -- Sta. Cruz de Tenerife, Spain

jbetanco@ull.es

Alejandro J. Castro

Dep. de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez s/n, 38271 La Laguna -- Sta. Cruz de Tenerife, Spain

ajcastro@ull.es

Lourdes Rodríguez-Mesa

Dep. de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez s/n, 38271 La Laguna -- Sta. Cruz de Tenerife, Spain

lrguez@ull.es

J. J. Betancor, A. J. Castro, L. Rodríguez-Mesa. “Characterization of Uniformly Convex and Smooth Banach Spaces by Using Carleson Measures in Bessel Settings.” Journal of Convex Analysis 20 (2013), No. 3, 763–811.