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

TL;DR

This paper characterizes Banach-valued BMO functions and UMD Banach spaces using Bessel convolutions and Carleson conditions, establishing unique inequalities for UMD spaces related to Bessel-Poisson integrals.

Contribution

It introduces a novel characterization of Banach-valued BMO functions via Bessel convolutions and identifies UMD spaces as the unique setting for specific Carleson inequalities.

Findings

Characterization of BMO_o(R,X) via Bessel convolutions and Carleson conditions

UMD Banach spaces are uniquely identified by certain gamma-radonifying Carleson inequalities

Establishment of connections between Bessel-Poisson integrals and Banach space properties

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 $\gamma$-radonifying norms. Also we prove that the UMD Banach spaces are the unique Banach spaces for which certain $\gamma$-radonifying Carleson inequalities for Bessel-Poisson integrals of $BMO_o(\mathbb{R},X)$ functions hold.