Factorization for hardy spaces and characterization for BMO spaces via commutators in the bessel setting

Fix γ > 0. Consider theHardy spaceH¹(R+, dmγ) in the sense of Coifman and Weiss, where R+ := (0,∞) and dmγ := x²γ dx with dx the Lebesgue measure. Also, consider the Bessel operators (equation presented) on R+. The Hardy spaces H¹δ γ and H¹ Sγ associated with δγ and Sγ are defined via the Riesz transforms Rδγ := ρx(δγ)-1/2 and RSγ := xγ ρxx-γ(Sγ)-1/2, respectively. It is known that H¹δ γ and H¹(R+, dmγ) coincide but that they are different from H¹ Sγ . In this article, we prove the following: (a) a weak factorization of H¹,(R+, dmγ) by using a bilinear form of the Riesz transform Rδγ , which implies the characterization of the BMO space associated with δγ via the commutators related to Rδγ ; (b) that the BMO space associated with Sγ cannot be characterized by commutators related to RSγ , which implies that H¹ Sγ does not have a weak factorization via a bilinear form of the Riesz transform RSγ.