Factorization for Hardy Spaces and Characterization for BMO Spaces via Commutators in the Bessel Setting

Fix lambda > 0. Consider theHardy space H-1 (R+, dm(lambda)) in the sense of Coifman and Weiss, where R+ := (0,infinity) and dm(lambda) := chi(2 lambda) dx with dx the Lebesgue measure. Also, consider the Bessel operators Delta lambda := -d(2)/dx(2) - 2 lambda/x d/dx, S-lambda := -d(2)/dx(2) +lambda(2) - lambda/x(2) on R+. The Hardy spaces H-Delta lambda(1) and H-S lambda(1) associated with Delta(lambda) and S-lambda are defined via the Riesz transforms R-Delta lambda := partial derivative x(Delta(lambda))(-1/2) and R-S lambda := x(lambda)partial derivative(x)x(-lambda)(S-lambda)(-1/2), respectively. It is known that H-Delta lambda(1) and H-1(R+, dm(lambda)) coincide but that they are different from H-S lambda(1) In this article, we prove the following: (a) a weak factorization of H-1(R+, dm(lambda)) by using a bilinear form of the Riesz transform R-Delta lambda, which implies the characterization of the BMO space associated with Delta(lambda) via the commutators related to R-Delta lambda; (b) that the BMO space associated with S-lambda cannot be characterized by commutators related to R-S lambda, which implies that H-S lambda(1) does not have a weak factorization via a bilinear form of the Riesz transform R-S lambda.