UMD-valued square functions associated with Bessel operators in Hardy and BMO spaces

TL;DR

This paper studies Banach-valued Hardy and BMO spaces in the Bessel setting, establishing norm equivalences and characterizations of UMD Banach spaces via square functions linked to Bessel-Poisson semigroups.

Contribution

It introduces new square function characterizations for UMD Banach spaces in the context of Bessel operators and Hardy/BMO spaces, extending classical harmonic analysis results.

Findings

Equivalent norms involving γ-radonifying operators and square functions are established.

Characterizations of UMD Banach spaces via Hardy and BMO boundedness of g-functions are proved.

The results connect Bessel operator analysis with Banach space geometry.

Abstract

We consider Banach valued Hardy and BMO spaces in the Bessel setting. Square functions associated with Poisson semigroups for Bessel operators are defined by using fractional derivatives. If B is a UMD Banach space we obtain for B-valued Hardy and BMO spaces equivalent norms involving $\gamma$-radonifying operators and square functions. We also establish characterizations of UMD Banach spaces by using Hardy and BMO-boundedness properties of g-functions associated to Bessel-Poisson semigroup.