Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions

Abstract

In this paper we consider the space \({{{BMO}_o(\mathbb{R}, X)}}\) of bounded mean oscillations and odd functions on \({{\mathbb{R}}}\) taking values in a UMD Banach space X. The functions in \({{{BMO}_o(\mathbb{R}, X)}}\) 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 \({{{BMO}_o(\mathbb{R}, X)}}\) functions hold.