Muckenhoupt-type weights in Bessel setting

Fix λ>-1/2 and λ≠0. Consider the Bessel operator (introduced by Muckenhoupt–Stein) ▵_λ:=- d^2/dx² - 2λ/x d/dx on R₊:=(0,∞) with dm_λ(x):=x^2λdx and dx the Lebesgue measure on R₊. In this paper, we study the Muckenhoupt-type weights in this Bessel setting along the line of Muckenhoupt–Stein and Andersen–Kerman. Besides, exploiting more properties of the weights A_p,λ introduced by Andersen–Kerman, we introduce a new class Ã_p,λ such that the Hardy–Littlewood maximal function is bounded on the weighted L_w^p space if and only if w is in Ã_p,λ. Moreover, along the line of Coifman–Rochberg–Weiss, we investigate the commutator [b,R_λ] with R_λ:=d/dx(▵λ)^-1/2 to be the Bessel Riesz transform. We show that for w∈A_p,λ, the commutator [b,R_λ] is bounded on weighted L_w^p if and only if b is in the BMO space associated with ▵_λ.