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Fix $\lambda>-1/2$ and $\lambda \not=0$. Consider the Bessel operator(introduced by Muckenhoupt--Stein)$\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ on$\mathbb{R_+}:=(0,\infty)$ with $dm_\lambda(x):=x^{2\lambda}dx$ and $dx$ theLebesgue measure on $\mathbb{R_+}$. In this paper, we study theMuckenhoupt-type weights which reveal the intrinsic structure in this Besselsetting along the line of Muckenhoupt--Stein and Andersen--Kerman. Besides,exploiting more properties of the weights $A_{p,\lambda}$ introduced byAndersen--Kerman, we introduce a new class $\widetilde{A}_{p,\lambda}$ suchthat the Hardy--Littlewood maximal function is bounded on the weighted $L^p_w$space if and only if $w$ is in $\widetilde A_{p,\lambda}$. Moreover, along theline of Coifman--Rochberg--Weiss, we investigate the commutator $[b,R_\lambda]$with $R_\lambda:=\frac{d}{dx}(\triangle_\lambda)^{-\frac{1}{2}}$ to be theBessel Riesz transform. We show that for $w\in A_{p,\lambda}$, the commutator$[b, R_\lambda]$ is bounded on weighted $L^p_w$ if and only if $b$ is in theBMO space associated with $\triangle_\lambda$.

Muckenhoupt-type weights and the intrinsic structure in Bessel Setting