arXiv2023
Fix $λ>-1/2$ and $λ\not=0$. Consider the Bessel operator (introduced by Muckenhoupt--Stein) $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ on $\mathbb{R_+}:=(0,\infty)$ with $dm_λ(x):=x^{2λ}dx$ and $dx$ the Lebesgue measure on $\mathbb{R_+}$. In this paper, we study the Muckenhoupt-type weights which reveal the intrinsic structure 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 $\widetilde{A}_{p,λ}$ such that the Hardy--Littlewood maximal function is bounded on the weighted $L^p_w$ space if and only if $w$ is in $\widetilde A_{p,λ}$. Moreover, along the line of Coifman--Rochberg--Weiss, we investigate the commutator $[b,R_λ]$ with $R_λ:=\frac{d}{dx}(\triangle_λ)^{-\frac{1}{2}}$ to be the Bessel Riesz transform. We show that for $w\in A_{p,λ}$, the commutator $[b, R_λ]$ is bounded on weighted $L^p_w$ if and only if $b$ is in the BMO space associated with $\triangle_λ$.