UMD-Valued Square Functions Associated with Bessel Operators in Hardy and BMO Spaces

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 $${\mathbb{B}}$$ is a UMD Banach space we obtain for $${\mathbb{B}}$$ -valued Hardy and BMO spaces equivalent norms involving γ-radonifying operators and square functions. We also establish characterizations of UMD Banach spaces by means of Hardy and BMO-boundedness properties of g-functions associated to Bessel–Poisson semigroup.