In this paper we consider the space BMOo(R,X) of bounded mean
oscillations and odd functions on R taking values in a UMD Banach
space X. The functions in BMOo(R,X) are characterized by Carleson
type conditions involving Bessel convolutions and γ-radonifying norms.
Also we prove that the UMD Banach spaces are the unique Banach spaces for which
certain γ-radonifying Carleson inequalities for Bessel-Poisson integrals
of BMOo(R,X) functions hold.Comment: 29 page