Muckenhoupt-Type Weights in Bessel Setting

Fix λ > - 1 / 2 and λ ≠ 0 . Consider the Bessel operator (introduced by Muckenhoupt–Stein) ▵ λ : = - d 2 d x 2 - 2 λ x d dx on R + : = ( 0 , ∞ ) with d m λ ( x ) : = x 2 λ d x 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 A ~ 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 A ~ 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 .