Square roots of the Bessel operators and the related Littlewood-Paley estimates

Let Δ_λ and S_λ, λ∈R₊:=(0,+∞), be the two Bessel operators studied by Muckenhoupt–Stein (1965). We prove that the square root of the Bessel operators and the corresponding “gradient” operators are equivalent in Lp spaces for 1<p<∞. Moreover, by using holomorphic functional calculus, we establish the characterizations of boundedness on Lp spaces associated with Bessel operators in terms of the Littlewood–Paley g-function with respect to the square root of the Bessel operator. Also, we study boundedness properties of Littlewood–Paley g-function associated with the square root of the Bessel operator on the odd BMO space BMO₊ and the atomic Hardy space H¹.